A sovereign nation is defined by two criteria:
1. Maintained territorial integrity.
1.1. The easiest way to establish unchallenged territorial integrity is to find a place in international waters where nothing is taken away from a specific nation. The logistical and territorial disadvantages of that approach are obvious though.
1.2. The best route to establishing territorial integrity on land is to buy territory. Land has been traded between nations in modern times (Alaska). Your best bet would be to approach an absolute monarch with money trouble and aim at land which does not have any apparent value (bummer!). If the oil prices continue to fall the Arab Emirates would be such a candidate. You may need to grant ongoing resource exploitation rights to the original owner though as a condition for the trade. But overall this should be one of the few ways to peacefully establish territorial sovereignty.
1.3. I'm not sure whether this qualifies, but you can hijack a country. This resembles the United States' South America policy in the second half of the 20th century (Chile, El Salvador). In order to prevent international intervention you should choose a country without resources or geo-strategic value (bummer!) that nobody is really interested in. You bribe the military into a coup and install your own people as government. Your main problem will be that the military will have final say in all matters; you will have to bribe them more than your competitors into supporting you long enough that you can install your own officers and eventually reduce their power and influence without creating their opposition.
1.4. Taking land away from somebody is a lot harder. In order to get an impression of what this entails we can look at Israel, a synthetic state established about 70 years ago. The neighbors weren't happy and the displaced original population is still hostile three generations later, even though they never set foot on their ancestors' land and even though the country is mostly desert devoid of resources. Israel had to fight a major war to defend its existence. It would not be here today without its own strong army and its ally, the U.S. Takeaway for land-based new nations:
- You need an army of a strength matching that of your combined neighbors.
- You need a strong ally, best would be one of the nuclear powers.
2. International recognition.
This is what my "ah, the sovereign state" comment aimed at. Your sovereignty hinges on formal recognition by other countries, preferably including those which have the determination and means to defend you. Formal recognition includes exchange of ambassadors, recognition of IDs and possibly a vote for inclusion into the U.N. The more of these criteria are met, and by more countries, the better. There is a wide grey spectrum though: Often a country is only recognized by its immediate allies but not by any other countries; their sovereignty is questionable.
Other indicators are often only symbolic to the point of being illusionary. For example, just the fact that nobody has invaded your apartment or farm in a while does not make it a sovereign territory. Issuing passports has no significance whatsoever unless they are recognized by other countries.
As a rule of thumb your country will be recognized by other nations if they have an advantage from doing so; and the complement is also true: Countries for which your recognition would be a disadvantage will try everything to prevent it. (Examples are West Germany not recognizing East Germany, mainland China and Taiwan, the two Cypruses, and Israel fighting the recognition of the Palestinian territories.)
Sovereign nation cookbook
If you don't find anyone to buy territory from:
My best bet to establish a sovereign country on land would be to hijack a warlord's organization in one of the failed African states (which are for historical reasons weak and awkward agglomerations with low cohesion anyway), find a suitable territory that has no significant mineral resources (bummer!), bribe the national government (which shouldn't be hard) into tolerating you and establish a peaceful rule including your own tax administration which taxes really little (because you have money already). Because your place is stable and works really well (as opposed to its surroundings) and you provide real good internet (you f-cking own the internet satellites, right?) you lure tax-savvy financial investors into your territory. Chances are you are close to the equator; then you could provide a rocket launch site, perhaps starting with the ten launches per day for your own internet satellites. Since you are the cheapest launch option this would make a lot of people and governments semi-dependent on you. This way you build an influencer group who supports and lobbies for your quest to become fully sovereign, which essentially at this point means to be recognized by other people. After a few decades of running the place successfully, on the way defending it against rogue generals and the occasional competing warlord, other nations will perhaps start to recognize you in order to be on your list of countries eligible for favorable business conditions, and your grandchildren or great-grandchildren may eventually send the first envoy to the U.N.