Incoming wall of text!
TL;DR version: Personally, I like 3 and 4, with 5 probably being my least favorite.
2 and 3 are probably the ones that are the most 'personal' In their stories. In 2 you have your mentor who you travel with/work with for the first portion of the game-which was rather effective IMO for setting up an emotional investment when things go south. I think that a lot of that is wasted because when things finally do hit the fan, and you confront the one responsible...you are likely far too underleveled to have a chance . As a player who got their pro-shaper loyalties riled up by the event in question, I felt incredibly disappointed in how there was essentially no real pure-shaper narrative route to take. Even with the shaper loyalist camp rather nearby, there just isn't a pro-shaper questline/faction to join, so you pretty much get railroaded into joining one of the factions that screwed you over/deceived you.
3 Is pretty personal in that it's the only one in the game that you can have a companion with you through the entire game (and two for most of it) and sets up the direness of the situation pretty much immediately with the attack on the school, showing off the main antagonist and your allies for much of the game. It could be rather linear at times, and I don't think I have ever met someone who appreciated the boat travel mechanic, but I think the stakes were pretty well laid out and dire in this game right from the start, and it conveyed the feeling of shifting lines of battle rather well IMO. The party banter was appreciated (and something that didn't pop up again until the Avadon games), if a bit inconsequential in the long run (and Greta says some really dumb things in some of the conversations) Out of all the games, I'd say I probably say that I felt the most attachment to the geneforge 3 protagonist, probably because so many of the characters pivotal to the plot are ones that the protagonist has a rather personal relationship with (alawan, Greta, Litalia, Hodge, etc) which isn't true to the same extent In the other games. A great candidate for a remake that deals with some of it's issues IMO.
4 is my favorite. Despite starting as a Rebel for a change instead of a Shaper, 'playing against the grain'-working for the 'opposing' faction never made me feel like I was acting against character, unlike in Geneforge II+III. It was nice to see some returning faces like Greta and Alawan etc, and the game really conveyed the feeling of an evolving conflict as your progressed through the game. despite being a person who used a geneforge by default, your use/non-use of cannisters has a huge impact in the game, particularly in the epilogue. Likewise, many actions you take influence the ending in rather substantial ways, and despite there being only two (plus one hidden) factions the endings were quite diverse, and you really felt the impact of your decisions. If I had to make any complaints about this great game, it would be that I wish Litalia had a bigger role in it-she seems like almost a non-entity in this game. Also, I liked the way Vogel kept the alignment of the Geneforge III protagonist nebulous through most of the game-it was a shame that it got thrown out in the last hour or so of the game.
5 was a bit of a disappointment IMO. A return to the static battle lines paradigm of the earlier games, going back to the first game in that using canisters has zero effect on the endgame. ( I know, I did a no-canister run-it affects perhaps 2 conversations), the ability to choose between Shaper and Rebel classes-which has no effect on the story, an amnesia subplot that gets dropped and never really satisfactorily resolved, and has you playing as a deformed enslaved mutant for most of the game-which is really jarring if you plan to do a shaper run-and IMO this games narrative *really* acts like it doesn't want you to play as a Shaper-there are only two Shaper factions...and encountered later In the game...and one of them seems to be Nazi inspired, complete with death camps, while the Rebels get the only factions that could possibly be described as 'moderate'. The endings are the same for a given faction pretty much no matter what you do up until joining a faction, the only real thing I noticed that had an impact was if you killed a character or not in the final battle, whose repercussions don't make much sense IMO. Rather disappointing I think, despite the graphical upgrade.
The first one is the only one I haven't beaten.