It's easier for me just to list all of my favorites from each type than to try and pick one to three from each, so here we go!! No duplicates on the list, even though some could fit into two categories.
Venasaur, thanks to Bulbasaur Propaganda + The mega, it became my fav Grass-type.
Arcanine, used it multiple times in competitive and playthroughs and I've always loved the lines.
Primarina, I used Popplio twice for different playthroughs and I've loved it for its cuteness.
Furret and Eevee are tied, but since I have Sylveon put Furret. Furret thanks to the memes I have loves its cuteness.
Tapu Koko, an electric chicken, I love it.
Gardevoir, I have used it a lot in Hoenn and grew attached to it.
Infernape, so versatile and fun to use playthrough or tutorial.
Lycanroc, Tough Claws + Acceleroc.
Swampert, I liek Mudkips so I love Swampert.
Dragonite... YOU ALL KNOW BY NOW!
Herracross, Always used it in Johto and so fun to hit stuff with Megahorn and Close Combat
Nidoking, such a beast of a mon and so fun to use.
Zorua, I think Zorua is very underrated in the cute department and I even made an OC of a Zorua which is fun to play.
Chandelure, It is a badass type combo with a monster special attack and awesome design.
Alolan Ninetales, Easily my fav Alolan form and I feel it is somewhat versatile with options for Aurora Viel, Blizzard, Nasty Plot, and other stuff.
Jirachi, FLINCH FOR DAYS!
Latias, aside from Noodles, I find it is a great Dragon-type, it and Latios, both are fun to use.
Sylveon, grew attached through Pokemon X and it is so adorable and I love it.