Personally, I think it's much easier to remain independent, and then once you build yourself up enough, you can consider forceful takeover of other provinces.
As a vassal republic, a huge chunk of income will be paid to your top liege. This on top of the gold you'll have to pay to your family members, will really seriously impede your progress of upgrading your cities/family home/trade posts. Gone are the days when there are no tradepost limits and no family member maintenance, where you can easily hit +50 ducats per month within the first few decades of playing a Republic.
A merchant republic's main strength is not their gold income, but rather their ability to field huge retinues once they have enough upgraded trade posts. If you are of Italian culture, you'll be able to field stacks of pure pikemen, which is one of the best retinues in the game. Due to their ability to field huge retinues without having to necessarily blob like crazy, I think Republics are currently the second strongest government type after Nomad. Republics are clearly the king of tall playstyles in CK2.
The weakness of Republics is that it is much harder for them to expand, and the power they obtain from expanding is very minuscule. Still, this is a paradox game, so at some point, you're going to have nothing to do but to expand. Seizing provinces via seize-city->seize county is a very slow and tedious process. Fabricating claims cost godly sums of ducats, since the fabricate claim cost scales with your income.
This leaves only inherited claims and holy wars. If Catholic, you can play the Pope game and try to get your dynasty elected Pope so they give you some free cbs.
Personally, I find converting to a heretical religion and then holy warring the simplest and fastest way to proceed (I find this the same even when playing as Feudal). You'll have to juggle this with the risk of getting holy warred before you're ready though. It is probably easiest to do this playing as someone like Venice who doesn't technically border anyone can defend its capital as the island.
