First off, a belated hello -- I'm a baby ME gamer and poster. I've only played the trilogy through twice so far, but have been completely in love with the series and will replay it many times to come.
One of my favorite aspects on my first replay of all 3 games was mixing up squadmates not just to test powerful combinations in combat, but even more importantly to simply see how certain combinations played out (and because I'm a sucker for dialogue and relationship development).
One of my favorite and most surprising combinations in ME2 was taking along Thane and Kasumi. Thane's rather gloomy outlook and dry humor was perfectly offset by Kasumi's light, witty humor and together they were incredibly fun. They ended up being so much fun that I started taking them with me on all my missions, whether suitable to the situation or not, just to hear what they might say. To go recruit Samara and the two of them start bantering with the cops about how many times a day my Shep endangers their lives, for instance, was just adorable, especially as at that point in the game it's already established that my Femshep is someone Thane loves -- and that Kasumi is aware of that. The funny thing is, I played this scene through with other squadmates whose scripts are nearly identical but they aren't nearly as funny. The delivery is everything.
(And of course nothing beats Thane/Kasumi when you meet the Biotic God. Thane's quietly witty, "Sleep well, and may you dream of bulbous women" is not only a delightful Princess Bride homage, it's just really really funny and keeps his character surprising and not the monkish stick in the mud he could so easily become.)
My favorite thing was unlikely combos, but at the same time, this also resulted in some real surprises. My first runthrough, I had Thane as a rather rushed LI for my Femshep, but I didn't really pay attention to his strengths as a squadmate. I just wanted him to run through missions with my Femshep and say things in that Thanelike way. (I know. Sorry.) I do remember that I got very offended when he would yell at my Shep for shooting at him (I was, like, "What! You love me!").
But anyway, in the rerun, especially running through to the SM, all I can say is dayam. Thane is a fantastic squadmate, provided you pair him with the right people and powers. My first runthrough of the SM, I didn't lose anyone but with Garrus and Zaeed as squadmates (yes, I just picked who I thought would be best in a wartime situation), I struggled with each wave of the battle.
My second runthrough, I again picked Kasumi and Thane (the two 'weaker' characters I was most attached to, since I had an all-loyal squad and wanted everyone to live) -- and then the freaking collectors and possessed-uber-collector-guy just melted away when I used Kasumi on tech/initial defenses, then biotic combos on them all with Thane (my character was an Adept). Then I basically let Thane snipe the remainders with assists from Kasumi and boom, done. It was insane how much easier that run was. The first run, it took me minutes at times to pick off the possessed collectors. This next time -- with "weak" squadmates it took me seconds--warpwarpboom, ashes. The evil collector guy didn't even get to finish his blah-blah-blah-reaper-fear-me dialogue. It really illustrated to me how much we can learn on each playthrough, too.
My first time playing, I couldn't imagine playing on insanity. Now I'm already planning an Insanity run in my head for the future.
I also loved trying to match backstories with situations. I was often unexpectedly touched by squadmate observations in this way -- for instance, I'd sometimes just take Mordin and Grunt for an opposites-attract run, or Tali and Liara, just to bring the old ME1 girls' group back together (and the voice actresses really played that sense of teamwork beautifully). Tali/Garrus is also fun back on the Citadel, as are Tali/Garrus and Ashley.
For me, I care most about character, then plot, then combat strategy (so I'll never ever be super-uber at this, and if I do ever do MP, you have been warned). I want to feel like these are real characters, so when you pair Thane and Samara places (another interesting combo, although Samara is so somber she makes Thane seem actually outgoing), you're not just pairing potential combatants, they're two sides of the same coin -- the warrior monk/warrior nun, etc. Or taking Samara on Jack's mission and seeing her quiet horror at what she views.(The only time this didn't work for me was Miranda/Jacob, who were surprisingly dull as teammates -- a shame because I love Miranda and was interested to see what we'd learn about Jacob (answer: nothing).)
Those kinds of character combos are so rich to me because they help to add texture to the universe my character lives in. I've played two femsheps so far -- one Paragon, one Paragade, and it was incredibly fun, but the second run-through was more fun because I paid attention to the supporting characters and their observations, a great mixture of joyful and sad and funny and grim. I know many don't love ME2, but for me it remains the richest and most rewarding ME thus far, because I grow to love the squadmates so much.