In my experience, the best way to get other people to try the Ace Attorney series is not to emphasize its fantastic writing, characters, and problem solving...
...but to instead draw attention to just how funny it is.
Ace Attorney is one of the funniest game series you'll ever play. It won't necessarily make you laugh super hard all the time or anything, but all the laughter you'll get from the situations and words the characters say will be 100% genuine. The series' sharp wit has a very distinct identity shaped by how punchy certain jokes can be due to the way the game's text boxes are edited.
Did you ever play EarthBound? Remember how, in that game, text would have different speeds of crawling across the screen depending on the context, to draw attention to certain lines or to communicate a certain way of speaking? The way Ace Attorney operates its text boxes is like EarthBound taken even further beyond. LOTS of care is taken to making its already sharp writing hit you as hard as possible with the VERY precise and exact way the text shows up, in every scene in the game.
Of course, the game isn't "on" 100% of the time with its text boxes, so the punchiness never gets exhausting. As funny as Ace Attorney is, it's surprisingly restrained at times.
So, yeah. If you want a game series that will make you laugh, please check the Ace Attorney games out. What other Ace Attorney fans have said is accurate and true, but I wanted to draw attention to its comedy and how it's done in a very special and considered way. I hope this helps put the series onto your "I gotta play this soon instead of later" brain category, haha.