Yes, you always get 2 grass, 2 trees and 2 sea. For this reason, if you aren't sure you should just settle in place. Moving your settler always involves some risk. For my money, being next to a hill isn't reason enough to move. The hill can be a disadvantage if you get rushed fast, but you can also put your own units on the hill to counterattack. You may also get a workshop later and get extra production because of the hill. The English and Spanish get production bonuses from hills later in the game. Don't fear hills!
As for moving, there are several advantages you can gain by doing so, but it always involves risk. Moving your capital in a MP game with Zulu or Aztecs in it is particularly dangerous as you may move closer to your enemy who might have a warrior to your cap before you can possibly defend it or the turn after you send your first warrior in the opposite direction.
On the positive side, you might move, pick up some gold on the way from friendly villages or exploration tiles and plop down next to somebody else's cap with 25+ gold to mount an early offense. You might wind up with some good resources in your capital, but keep in mind this is only a great advantage if you can use the resources right away. Otherwise you could just settle another city on that resource later.
Egyptians often move to try to get more deserts. This is risky and I sometimes see Egyptian players settling in pretty bad places. Even if you get your deserts, if you don't have grasslands, you'll grow very slowly and that'll hurt in the long run. Settling without trees is pretty bad as you could be defenseless when an enemy finds you or at the very least be quite slow to get warriors and miss out on barbs.
Spanish move a lot too. Personally, I don't like to do this because it's risky and I don't think the benefits are that great if you can get a settler quite fast. Just put your settler on that whale. But some players have had great success moving Spanish settlers. The reason this works is because the Spanish get double gold for exploration tiles, so if you just hit one, you'll settle with 20 gold and can rush a warrior immediately. If you settle by 3700 BC, you haven't really lost anything.
Greeks should usually move one tile in any direction as this will almost always reveal a resource in courthouse range. You still settle in 4000 BC. But beware, you really want to have four grasslands and four trees with the Greeks, but are only guarenteed to have two near your starting position. If you settle on one of those, you might begin the game with only one grasslands or one tree and that's really going to hurt your development.
The Russians are my favorite civ to move with because you can beeline a friendly village to pick up 25 gold or a tech and then pick an ideal spot to settle.
I don't put much stock in moving with any of the other civs. I see people do it with the English from time to time (to find dye) and that never has worked out. I've occasionally moved one tile with the Germans just to trade three grasslands for three trees or something like that.
It's something fun to try for a change of pace, but there's definitely some skill involved in picking the right spot. There may or may not be a better place to settler, but I guarentee there are always lots of worse places. Above all, don't move too long. If you don't find what you're looking for in the first couple turns, you're just going to have to pick a spot and make the best of it.
