The first thing to understand is that fins are not strictly necessary snorkeling gear; I've often snorkeled without them. Fins compensate for not using hands while swimming. If you have an underwater camera and/or anticipate strong currents, you'll want to use fins in order to help you propel yourself along under water. Even if these don't apply, having fins makes the experience more comfortable since you can let your arms dangle by your side without sacrificing much mobility.
All other things equal, bigger fins are more efficient in the water and more of a hassle out of the water. It's possible to buy stubby fins designed for excercise and packing in carry-ons. On the other extreme, there are monofins used by freedivers. Unless you have specialized needs, just go for the traditional sizes which roughly double the length of your foot and aren't excessively wide. These provide more than enough thrust and don't require learning new swimming techniques. You're already committed to using some luggage space for the mask and snorkel; regular fins don't demand much more.
Growing up, my parents bought us boys fins with straps (i.e. open foot). The main advantage was that our feet were growing so fast we'd change sizes from one trip to the next. Straps, however, are fiddly. When I bought my own fins as an adult, I picked closed foot (full-footed) fins that fit my fully grown feet. As long as I remember to get both feet and fins wet before pulling them on, full-foot, rubber fins are a lot more comfortable. I also learned that open-foot fins are intended to be worn with booties for insulation or to protect your feet while entering the water. For most tropical locations, you won't need extra protection unless the shore is particularly rocky and rough.
I've never tried split fins, but I understand that good ones are more efficient than traditional paddle fins. For my money, I don't really need more propulsion than I get from my current set. But I'm considering investing in a pair for my wife since she isn't as strong a swimmer as I am. According to this guide:
Your leg stroke with a split fin is very different. Instead of long forceful strokes, you make much smaller, easier, and more rapid kicks in a split fin (sometimes called a flutter kick). You feel less resistance with this type of stroke, through the water, and through your fins.
The things folks don't like about split fins is that they don't provide as much feedback, because you don't push against the water as hard with them, and they are considered to be less precise for control of movement and position in the water. They are not supposed to be as good for back paddling, frog kicking, turning, etc. They are better for moving forward. But we have had no problems with them.
Finally, make sure fins fit. No matter how great a pair of fins are, they aren't much use if they don't feel comfortable in the water. Since your feet will shrink a bit when they get cold, prefer slightly too tight in the store to slightly too loose. Check the return policy of whichever retailer you plan to buy from and don't hesitate to exchange if you don't enjoy your first outing.
