Obviously the smaller pieces are generally easier to get rid of than the larger pieces, but I'd be interested to know which pieces I should aim to get rid of first, and which pieces I should save. I'm sure this largely depends on the gaps on the playing board, but are there some pieces that go better in the majority of cases?
For scoring purposes alone placing the pentominoes first is the optimal approach.
What you place first depends on your strategy. Sometimes I like to go straight after another player to attempt to block them in. Y, I, and L are good opening pieces for that. For a more balanced to the middle approach, I like to open with F or W. They give lots of corners to build off of and are not immediately threatening to the other players.
I, P, T and X can be difficult to place later in the game, so I like to get them out as soon as possible. F, U, L, W and Z are all great for cutting other players off, so I like to save them for when I get close to another color.
Usually only 2 of the 4's give me trouble; the straight piece, and the perfect square. I tend to prioritizing placing them over all the smaller pieces and even some of the 5's.
In most games what you place becomes highly dependent on what the other players are doing. Especially in later turns. You need to give yourself room to grow while simultaneously denying space to your opponents.
Most certainly; all pieces are NOT created equal, some pieces are greater than others.
Of course, none of that takes into account how easy or difficult a piece can be to play, merely how strong it can be after it's been played. Additionally, the situation can make a piece much stronger; the long bars are excellent for making a long jump into enemy territory, and if they can be played to block an opponents expansion into that area then they're the perfect piece for the moment. In general though, the vertex count gives me a good and easy way to rate pieces.