To start I really enjoyed Paper Mario, and I really look forward to further games in the series. The game play was really enjoyable for me.
Fighting - Some people don't like turn based fighting and prefer more action based mobile fighting style. I enjoy both but especially in this game turn based just seems a bit more fun. In the beginning Mario could only do one hit with either his jump or hammer attacks but over the course of the game he was able to do at least two jumps and stronger strikes of his hammer, and the commands were more interactive and required timing which made it a bit more exciting in battle. Then counting the various companions that Mario picks up along the way, item attacks, and Star Powers, there are countless combinations to use in battle to quickly dispatch those rogue Koopas or Lakitus. So in short I really enjoy the battle style and correct me if I am wrong but this was the first Mario game to use such battle style and it was then used in games such as the later Paper Mario games and the Mario & Luigi series.
Badges - The Badge system is pretty fun, and adds more to the game. Badges are items that can be found, bought, or are gifts from various NPCs; that add special moves or stats to Mario. One of Mario's stats he can level up per level is BP or badge points, each badge takes a certain amount of BP to equip. The fun thing about leveling up BP more than health or FP (what is used for special moves) that there are badges that can add more HP or FP. Badges also can increase the amount of damage Mario does on regular attacks or decreases damage taken. Move badges range from just stronger Jump attacks to Jump attacks that can continue until the command is missed, or a hammer charge that makes the next hammer attack stronger.
My favorite two badges are Zap Tap and Mega Quake. The Zap Tap badge makes it where physical attacks on Mario damage the enemy for one hit, which is fun with Fuzzies the enemies that attach to Mario and suck HP but what is fun is Zap Tap stops that and Fuzzies can't even touch Mario. Mega quake and Quake badges in general are fun because it allows a hammer move that hits all enemies on the floor and roof.
Star Pieces - Star Pieces are collectable items in the game scattered throughout. They can be in trees or underwater or a prize from a quest. The most prevalent of quests is the mail one with Parakarry, when you first meet him you find three letters and can find more in various places, when you deliver them you either receive a Star Piece or another letter and continue the chain. Other quests are the quests of Koopa Koot the elder of Koopa Village he sends you on errands and normally rewards you with the fantastic amount of one coin, but sometimes he gives you three whole Star Pieces. Star Pieces can be traded for Badges but I never really did that I just enjoyed collecting them and spent quite a lot of my game time on it.
There are many other awesome features of the game that are not coming to mind, but I hope if you ever have the pleasure to play it you can explore everything.