I can't be objective about this, I saw it so many times as a kid - and what's more, recited key pieces of dialogue, and had dialogue recited to m,e by friends. Some of it hasn't aged well - Mahoney sexually harrasses lots of people, the dialogue is littered with casual homophobia, it has a lot of 80s comedy moments (random topless women at parties) and many jokes do not hold up (eg the lecherous fake Italian making racist comments to a Japanese woman during the riot... isn't he supposed to be in love with Callahan by then? A lot of characters feel under-developed eg the henpecked husband (he has a great intro sequence then is kind of forgotten), the rich girl played by Kim Cattral (I kept expecting her rich mother to come back).
But it's beautifully shot, the score is memorable, I really liked the location they got for the police academy. Dramatically it's simple but very effective - within the first ten minutes they set up the basic situation: the police academy are accepting everyone and it really annoys the police... and the hero Mahoney is forced to become a police officer or he'll be arrested. Now that's good basic solid dramatic conflict - it gives Mahoney a goal (get kicked out) and an arc (to become a good cop); it motivates the baddies (to get rid of these recruits). There's a solid second act end point with Hightower and Mahoney getting kicked out
There are some excellent gags, like two bullies throwing some books out a window only to find the window is shut, and the wife running through backyards to catch her husband.
It's also extremely well cast. Steve Guttenberg is charming for all his sexual harassment; Kim Cattral is very sweet and teams well with Guttenberg; George W Bailey is an excellent villain (well supported by George Robertson, Brant Van Hoffman and Scott Thomson); Michael Winslow is great fun (a different kind of comic too - he gave the series a real point of difference); David Graf is hilarious in probably the series' best role; George Gaynes is hilarious as the ditzy commandant; Leslie Easterbook is fun as the well endowed Chapman; Bubba Smith is interesting looking and does provide some moral ballast when he sticks up for hooks on a race basis; Marion Ramsey is very relatable as Hooks; Bruce Mahler is well cast as the nerdy guy (I wish they'd used his wife more). Andrew Rubins' ladies' man and Donovan Scott's fat man are fine - though it was no loss they didn't return for other films in the series. (I did like Scott's arc where he got to beat up the bullies who taunted him at the beginning.)
This is more raunchy than later films in the series with some nudity, and gags like Bailey flying in to the back of a horse, and Georgina Spelvin performing oral sex on Gaynes and then Gutteberg (a nice call back).