BEWARE THE WALL OF TEXT
The best was, without a doubt or any competition, Sonic Advance 2. There are lots of analysts out there who have shared their opinions on the Advance series, but I think the fastest, most effective one was Youtuber
ShayMay.
To begin with Advance 1 and 3 have some fairly prominent problems. Sonic Advance 1, in my opinion, was bland from start to finish. The real problem with it, though, was the stage design. Since you'll be punished for disobeying the level designers' rules, the game is very stifling. To me, this applies to all levels outside of the first one, which is just a few straight lines and bridges with no imagination or compelling design. Yet, in Egg Rocket, the final proper stage of the game, you're punished for actually following the level designers' rules. For a couple of examples, there's a speed booster that forces you to back-track in the level and a leap of faith over spikes that you must take in order to continue.
Sonic Advance 3 suffers from it's own problems as well. My main problem with the game was the maze-like stage design it could sometimes have. I ended up backtracking multiple times in Sunset Hill, Ocean Base, and Cyber Track in ways that couldn't be solved by simply flying back up. Again, the way the stages are designed communicates that you must follow the designers' instructions at all times, which feels suffocating. And that's to say nothing about the arbitrary puzzle solving and button pressing or the nearly pointless team mechanics that require incredibly specific scenarios to even be relevant.
Sonic Advance 2, in my opinion, is not just a better game, it's the only one I'd say is an excellent game. The boost mode is probably "the greatest addition to the core Sonic formula...since the spindash", and it came right from this game. "Unlike the later versions where you could just press a button to activate it," the original boost mode required the player to do a substantial amount of running. The time it takes to activate decreases depending on the amount of rings you collect, encouraging you to, and rewarding you for, running and collecting rings which, obviously, are pretty important to Sonic. Unlike Adv. 3, though, Adv. 2 has level design that actually compliments this mechanic. In Adv. 3, the speed sections were incongruous with the maze, platforming, and other sections of the game, and so had to exist in a vacuum. In Advance 2, speed, platforming, and effective usage of extra controls such as the trick system and character-specific abilities (Sonic's airdash, Cream's flight) all flow into one another in a really "fluid" experience, and the challenge of maintaining your speed made the gameplay compelling yet never frustrating as it was optional. The only part of the game that really seems incongruous with the rest is the exploration, which is encouraged through the SP rings that will lead to a special stage if 7 are collected. Though this slows down the pace of the game, that in itself isn't a problem as it's the player's choice. However, the fact that it's often impossible to back-track far enough to get one you missed is a pretty big problem as it defeats the purpose of slowing down and exploring entirely.
The SP rings, then, have led to a lot of playthroughs of the game misrepresenting what it actually looks and feels like to play it. The best way to forge your opinion of it is to actually play it yourself. As usual the soundtrack is amazing, though this applies to the other two as well. The stages are very inspired thematically, especially Music Plant and Sky Canyon which stick to the themes through and through. The stage mechanics and set-pieces are well integrated and creative as well, such as the fans in Sky Canyon that allow the player to build up boost mode before continuing or the percussions the player must balance on if they want to take a different route. The most important part about the characters is that their abilities actually matter and affect both the way the game is designed and the way players will approach a situation depending on which character they chose. The trick system also promotes mastery of the game and replayability. Aside from that, it's a nice feature if you just want to look cool, and it gives you points when you pull them off. The special stages suck, but they still don't give me the perspective problems of 1 and 3 (or make me wanna barf for that matter). Of course, even within the stages themselves, the game is not perfect. There are still sucker punch moments in the game but they are so few and far between that I could count them on one hand. In conclusion, while I'd say all 3 titles are good pick-up-and-play games, 2 is the best choice as a Sonic game as well as the best example of one.