What are the arguments made in favor of a minority government over majority government and vice versa? I've heard majority is better because they typically last longer, but I don't see why this is necessarily a good thing (actually it would seem a bad thing as they can get away longer with doing something). Also I've heard minority governments have to work together more, to pass bills and such, but again I don't necessarily see how this is a good thing?
As you rightly mentioned in your question, arguments in favour of majority government is that they tend to last much longer, bringing with it an element of stability. Whether you support the party in power or not, a majority government brings with it a clear sense of what to expect for a particular mandate period. In the Canadian political system, some argue that majorities are not healthy because it lacks a system of checks and balances (like the US system) and gives the prime minister and his or her government virtually the power to do almost anything they want throughout their time in office. A united opposition would be unable to prevent government measures from passing. Good if you support the governing party; not so good if your political views or priorities are not aligned with the government.
Minority governments are interesting because they compel the governing party (which is usually the party with a plurality of seats in the House of Commons) to find common ground with one or more opposition parties. This forces upon elected officials to communicate and collaborate on the issues and to pass laws which, ideally, would benefit a larger swath of the population.
The downside to minority governments are many; however. First of all, with a shelf life averaging just over 2 years, there is a cost to the taxpayer in terms of the frequency of general elections. For example, since 2001, Canada has seen elections in 2004 (Martin minority), 2006 (Harper minority), 2008 (Harper minority), 2011 (Harper majority) and 2015 (Trudeau majority). Each election comes with a $375-million price-tag, with the 2015 election roughly double that amount as it was the longest election period in Canadian history.
The other downside is that minority governments often get defeated trying to accomplish the very things they set out to do in the electoral platform. Electing a minority government doesn't give the public a sense of predictability about what the government may do, because they require support from the opposition who come with their own demands. Political platforms are essentially invalid with a minority government either because they don't have the support to fulfill those promises, no time to complete them (being defeated before passing anything they've promised) or fulfilling part of their promises with some alterations proposed by opposition parties.
Another element to minority governments in recent Canadian elections is the changing of political party leaders. The Liberals have gone through 6 party leaders since 2006 (including interim leaders); The Progressive Conservatives went through this kind of phase after the Joe Clark government was defeated, and again when the Reform Party was established. When you combine party leadership ballots and the volatility of minority governments, it can really give someone the sense of a continuous, never-ending election cycle.
Which is better? It really all depends on what you value in our political system. Do you want collaboration over certainty? Do you want predictability or the power to remove an incompetent government?