Picking what to build where can really depend. I play TEC mostly, and when I run into desert world my usual inclination is either to turn them into an econ giant or a forge world with as many shipyards as i can cram into it. Of course, it varies based on your situation and map. If there's only one desert planet but it's at a great spot for attacking or defending, you might want lots of shipyards. If it's farther back, trade ports are probably better. I suppose you could put a ton of research stuff into it, but generally i'd think it would be better to spread those out to prevent a single attack from knocking you back to the stone age, so to speak.
Culture is another option, one which the ai occasionally seems to like. It's annoying, but since culture weakens as it passes thru each gravity well and divides itself up across phase lanes, your money is probably better spent dividing up the culture generators and going with one of the others, unless you're in a culture war for some reason. That said, I played a game against the ai where it pulled an all-culture desert world on me, and between that and having a half a dozen broadcast centers spread out across adjacent jumps, i ended up losing several gravity wells and almost losing a few more before i built enough of my own to counter them. As such, in some situations it's possible for culture to be a good choice, assuming the planet is well located and you get the element of surprise. Just make sure to capitalize on the situation before a) they build a bunch of their own culture centers or
they send a huge fleet to pull an Alderon on you... unless, of course, that's what you want them to do.
Overall though, culture is too slow to make much of a difference compared to either of the other two main options (loads of money or mass production - i imagine the vasari could do really well with a single centralized shipyard and some phase gates). The bonuses it gives are nice, but not earth shattering. Im not sure off the top of my head about the Advent or Vasari, but the TEC just get extra antimatter regen IIRC. As such, 9 time out of 10, extra logistics are best spent on a well planned trade route, particularly if you can route it through a desert world. Just make sure when you start building to have a clear idea of what your current territory is, your likely future territory will probably be, and plan ahead. Put research stations away from your best possible trade route where possible, if you're the TEC remember you can research to get an extra 4 spots at EVERY colonizable gravity well (even the dead asteroids), and if the game is likely to go on long enough, make sure to plan for the possibility of enemy culture on border worlds.