You are a player looking for a new game. You have just moved to a new city, so none of your gaming friends are available.
The "new city, no gaming friends" part changes my answer significantly.
While at a convention I'll game with people I won't see again, I've learned that it not worth getting together with people you don't like (not saying dislike, just little in common with) just to game. So I'd join some games at FLGS to get a feel for other players and GMs, and from there look to get invited to a home game with some people like (trusting them on the others). I don't think I'd be overburdened with too many choices that I could afford otherwise to be picky.
Let me flip it around: Dozens of friends are playing and I'm in the place to join a new game. What do I want to join.
First, given the option I'd prefer to get in at a campaign start rather than an ongoing campaign. Plot arcs around your character, get to learn them mechanically from 1st, understand all the background and recurring NPCs and foes, etc. That's not a deal-killer, but a strong point.
Second, I enjoy BOTH heavy RP with shades of grey and lots of combat. Both as in both at the same time, though combat is the more disposable of them - in a good campaign where we often will have multiple sessions between combats. On the other hand I find that I don't like campaigns where I need to piece together clues from two years ago RL time and a chance remark an NPC made last session, even though I run those.
Two years makes a good point - I like long running campaigns that are more serial than episodic, where I can make a change in the world (including by failure). I like games where the DM is up for more than one possible solution. I like games where there are multiple directions to go and the PCs need to prioritize and figure things out.
The actual setting is likely either neutral or a big plus. Heh. For example, most faux-medieval-european-with-elves-and-magic are interchangable to me, so neutral. Though something like the FR has so much stuff that my character should know and a DM might expect us to know without telling us, so that would be a negative. Actually, stuff where I as player can help flesh out is much nicer than things like FR. Something new (especially including homebrews that do shake things up) are usually a big plus, especially if the DM is good about presenting new knowledge the characters have but the players don't. My first time in Eberron was great! Dark Sun as well.
I prefer more serious games with some levity. Silly games don't keep my interest in a campaign though I've enjoyed one-shots or short-runs of Toon, Paranoia, TFOS and the like. I also like games with heavy RP (as mentioned) and a modicum of intra-party drama (with NO inter-PLAYER drama) to spice it up.
I like character customization, so options are good. The most memorable games are where the DM wants a particular feel, so adding/removing options as long as there's a theme is not only okay but encouraged. A good system lets people play the archetypes of the world - a great system encourages it.
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