I understand that the government collects taxes, sells bonds, etc. but
regardless of the income portion, I'm more curious about the spending
Does the US Treasury simply print the money (physically or digitally)?
Does the US Treasury have bank accounts/checking accounts?
What role does the Federal Reserve play in this?
Basically, I'm looking for the "School House Rock" portion on the
spending aspect (not the revenue aspect)?
The mechanics of how a payment is made are fairly easy and very similar to those of any other business.
The U.S. Treasury has accounts at banks that are part of the federal reserve system and write paper checks or makes electronic transfers out of them.
Theses accounts receive deposits from tax collections, proceeds of bond sales, land sales, fines in court cases, etc. Particular groups of bank accounts corresponding to line items are called "funds" and the Treasury Department moves money from one fund to another as required by law.
When an agency wants to spend money that it has been authorize to spend by Congress, basically, the agency fills out a form sent to the Treasury department saying that they want to write a check for $X that is authorized by this and that legislation and contract, and if everything is in order, the Treasury Department cuts the check. The process of determining that everything is in order is the hard part and adds meat to what would otherwise be a merely mechanical banking transaction.
An official in a public or private enterprise who keeps accounting controls in place to make sure that checks issued are authorized is called a "controller" or "comptroller" although the "comptroller of the currency" doesn't personally carry out this task any more than the "surgeon-general" personally conducts surgery on anyone in the course of his or her official duties.
Obviously, it is all more complicated than that with a lot of expenditures like payroll operated on a high automated mass produced basis, and authority is much more delegated than this simple explanation suggests, but that's the gist of it.
In practice, the Treasury Department puts money into funds with bank accounts for particular agencies or subagencies to use, based upon appropriation bills, and keeps track of who in an agency is authorized to state that it is O.K. to make a payment on a particular government contract because the government contractor has performed and the designated person, in turn, had minions who actually determine that the government contractor has performed who report to the designated person before the designated person signs off on a check. If not all of the money in a bank account for a fund is used at the end of the fiscal year, the Treasury Department decides what happens to the money (based on the relevant laws and regulations).
A related much more difficult question, which was not really asked and is not fully answered here is:
What is the procurement process by which they government authorizes
entering into a contract to buy something from a government contractor
pursuant to which the government is obligated to pay when the contract
This is a highly specialized area of administrative law (called government procurement law), and is very complicated.
It involves issuing "requests for proposals", getting bids on contracts and evaluating the bids to see if they meet the requirements, and then issuing long, complicated contracts and determining if they have been complied with by government contractors. It is the quintessential executive branch activity, much of which is buried in regulations rather than described in statutes.
One particularly complicated issue is how fine grained the purchases should be. Do you purchase a fully outfitted warship, for example, or do you buy your missiles and fuel separately from a different contractor?
Also, some procurement is authorized to be made via "no bid" contracts if it fits certain exceptions to the general rule that contracts must be bid, and price isn't the only consideration when accepting a bid. Administrative law also requires government contracts to comply with generalized requirements like paying a "prevailing wage" that makes them more expensive and supports union workers.
For further reading on government procurement in the United States, see here.
The complexity of government procurement in the U.S. is largely a response to the corruption that was used to benefit political allies and campaign supporters in a patronage system in the late 19th century and early 20th century in the U.S., before bid based government contracts were required, in parallel to the invention of a civil service system for hiring government employees to replace patronage hiring based upon political support for government officials who are employees.
How does printing money and coins figure into this system?
When the government makes coins or paper currency, money is spent from a Treasury Department bank account to buy the plant and material and labor to make the coins and paper currency.
Then, the coins and currency are distributed to banks (or a tiny percentage directly to private buyers who are coin and currency collectors) at face value in exchange for payment from the bank or other buyer through the federal reserve banking system.
Thus, on paper the U.S. Mint and the Bureau of Engraving and Printing makes a profit doing this which is equal to the cost of producing coins and paper currency respectively and its face value. This profit is called "seigniorage" and appears as a revenue item on the books of the United States government.
Seigniorage is a tiny percentage of the revenues of the United States government, about $30 billion a year (0.2% of GDP and 1-2% of federal revenues) as of 2013, out of overall federal revenues of roughly $3,490 billion as of 2019.
How is the money supply increased?
This is a tricky area of the economics of monetary policy and best asked in an economics forum for a more precise and complete answer.
In a nutshell, money is created every time anyone in the economy issues debt that is assignable to third parties, and is destroyed every time that anyone in the economy pays debt that had been assignable to third parties.
Basically, while we commonly think of "making money" as an exclusively governmental function, this isn't really true. Actually, almost everyone makes and/or destroys money almost every day. Every time you use your credit card, you make money. Every time you pay your credit card bill, you reduce the money supply. Government actually has a monopoly on defining what counts as money and regulating it, but economic equivalents to money are allowed, and are created and destroyed by every participant in every monetarily based economy every day.
Coins and currency are basically just transferrable loans made to the government on an interest free basis, but they are worth more than an interest free IOU from me because the federal government has a better credit rating than I do.
U.S. money, since 1971 when it left the gold standard (there were also periods of time when it was on a silver standard historically), is a fiat currency, and doesn't correspond to stockpiles of any physical commodity.
It is also important to note that only a modest percentage of the overall money supply is in the form of coins and paper currency which is called MB by U.S economists. The vast majority of it consists of accounting entries that do not have tangible form. Broader definitions of the money supply including M2, M2 and MZM. The chart below (from here) illustrates this fact:
As of 2011, M2 was about $10 trillion. M0 which is coins and currency in circulation (rather than in government vaults) as of 2016 was about $1.5 trillion. A large share of M0 is in the form of $100 bills in circulation outside the U.S. (about a third of it, which is about $500 billion) where it is used as a backup currency in places where the local currency is subject to hyperinflation or government devaluation risks, or the local government doesn't have a well functioning currency, or an a light, untraceable way to transfer money illicitly.
Who profits from making money?
Commercial banks that issue debt are a major creator of money and make lots of their profits by creating money although how this works is subtle and really takes an hour or so long lecture to really explain properly. The amount of money that a bank can create and profit from, however, is not unlimited. Primarily, it is a function of the reserve ratio (this is 10% in the U.S. to slightly oversimplify the situation), which is a banking rule imposed by the Federal Reserve, and FDIC and similar agencies (depending on the type of institution making the loans), that limits how much money a bank is allowed to lend when is has a certain amount of money deposited in an account at the Federal Reserve. Systems that operate in this way are said to be engaged in "Fractional-reserve banking."
Since the Federal Reserve banks in the United States (and almost all other banks) are owned by private shareholders, or are non-profits, rather than by the government, the profits that arise from fractional-reserve banking do not fund the U.S. government, they fund they owners of banks or the non-profits that operate them.
In contrast, the U.S. government does not directly make a profit from fractional-reserve banking which is the main way that they money supply is created.
The money supply has increased by about $200 billion a year over the last decade (per the link to M2 above). Other than the $30 billion or so a year of seigniorage that the federal government makes from printing coins and currency, the other $170 billion a year on average that came from making money that led to inflation and effectively taxed everyone to whom nominal dollar debts were owed like people with bank deposits (which are loans made by individuals to banks). This $170 billion a year, on average, of profits from making money was made by private parties, mostly by shareholders of commercial banks.
How does this relate to inflation?
Inflation is basically due to the amount of the money supply (broadly defined) chasing the amount of goods and services produced by the economy. As a first order approximation, the money supply (in a given currency unit) divided by GDP (in a non-monetized sense), governs the value of the unit of currency used. This is, of course, actually much more complicated, largely because creating money and destroying it isn't a centrally controlled process to the extent that most people think it is, and also because money supply and GDP aren't entirely independent of each other (partially for fundamental reasons, and partially because in a stable economy like the U.S., lots of obligations are defined in terms of nominal dollars rather than in terms of inflation adjusted dollars).
In countries like Weimar Germany and modern Venezuela, lots of commercial banking (and hence lots of the profits from increasing the money supply and thus creating inflation) are earned by the government owned central bank rather than private shareholders. So, the central government has an incentive to raise money through making money rather than imposing taxes (even though inflation is economically equivalent to a monetary wealth tax), since taxes are unpopular.
But, when production of goods and services is low, and the reliance on this means of generating government revenues is great, you get hyperinflation which utterly disrupts the economy and wrecks havoc on the monetary system. This is counterproductive, however, and often generates more unrest and dissatisfaction than raising taxes would have generated. A fractionally-backed banking system with fiat money relies on trust, reliance and confidence to work and if those are undermined the whole currency system ceases to work and people start resorting to alternative back up currencies that they do trust and can rely upon, like U.S. dollars or Euros, or commodities like grain or gold.