I'd echo everything Justin says other than UYA being the best in the series. A Crack in Time pretty much perfects and balances everything in the series really well. It has insane action, platforming, big open world sections and not only has a good story is about on par with UYA for its comedy. Clank also has the best Clank sections he's ever had with some fun Braid-esque time puzzles. They expertly keep the characters away from each other for such a huge chunk of the game it's a real "Fuck yeah!" moment when you're playing Ratchet and Clank again.
I've always been a little disappointed with Tools, and the remake is a good place to start but ultimately most of the complaints in that video are spot on. Nexus is also a really, really good game that both stands as a great epilogue to the PS3 games and a real love letter to the fans. I still can't believe Sony sent that game out to die by releasing it on the same day the PS4 launched.
Deadlocked/Gladiator is a good time, but they clearly took too many things to the extreme, almost eliminating platforming completely, but the weapons do all go to lvl99. The most unforgivable thing in that game is how much they sideline the series best character, Clank.
The 2D platforming bits in UYA were really novel at the time. There wasn't a barrage of nostalgic 2D platformers at the time, and it was a fun distraction. Now they're not so great (and even worse if you want the trophy) but the ghost pirate story that accompanies them is hilarious, and is finally paid off in the future games. UYA is a very, very funny game, mostly thanks to the combo of Lawrence/Nefarious. A lot of it is very specific to the time, but I've always been amused that they hint that Quark has had sex with a monkey in what a lot of people see as a kids series.
I kinda hate edgy Ratchet from R&C.
Don't play the PS2 ports of the handheld games.