Earthbound on SNES is a timeless one. Advance Wars is an excellent turn-based series. I played it on GBA, but there are a bunch of versions.

It’s a top-scrolling shooter, but give Tyrian 2000 a go. It’s free on GOG.

The device looks like it’ll be good to play most games upto the PSP. While I can suggest the usual suspects, Super Metroid, Metroid 4, SotN, Pokemon I’m going to suggest odd-ball games which I like

Game Boy

• Super Mario Bros Deluxe (NES Mario 1 & JP Mario 2 with saves and bonus features)
• Harry Potter 1,2,&3 (Turn Based RPGs, bit rough but an excellent sound track)
• Mickey’s Speedway USA (made by Rare, fantastic soundtrack, top down racer, GBC game so controls are wonky)

GBA

• Crash Nitro Kart (smoother and plays better IMHO than Super Circuit, controls require some patience to master)
• Spongebob Squarepants the Movie the Game (Made by Wayforward, and is better than I’d expect a license game to be)
• Jurassic Park Operation Genesis (Zoo tycoon with Dinos on GBA)

Genesis

• Ren & Stimpy Stimpy’s Invention (beautiful and painful platformer)
• TMNT Hyperstone Heist (Turtles in Time, but legally distinct so Nintendo didn’t sue)
• Scooby-Doo Mystery (A point and click adventure on the Genesis, yes they made one)

SNES

• Top Gear (It has a soundtrack that slaps)

PlayStation

• Pac-Man World 20th Anniversary (3D Platformer that’s aged well)
• Scooby Doo and the Cyber Chase (A 3D Platformer that hasn’t)
• Harry Potter 1&2 (We have the Legend of Zelda Ocarina of Time at home)
• Rugrats Search for Reptar (if you like the show you’ll like the game, otherwise it makes PS1 Hagrid look like the monalisa)
• The Emperor’s New Groove (3D Platformer that’s aged a lot better than it should’ve)

PSP

• Star Wars Battlefront II, Renegade Squadron, & Elite Squadron (same concept, 3 different implementations, all have their pros and cons)
• Modnation Racers (yes they ported a PS3 game to a portable with less power than a PS2, and it’s not half bad)
• LittleBigPlanet (same deal, just with a platformer)
• LEGO Star Wars II/LEGO Indiana Jones (PS2 games ported to PSP)

My RP3 is how I got completely absorbed into the trails series. Starting with Trails in the Sky and Trails in the Sky Second Chapter on PSP.

Cannot recommend Panzer Dragoon Saga enough. One of the best looking Sega Saturn games. Its a JRPG with turn-based battles, a super unique system. Along with the rest of the Panzer Dragoon series (though they are on-rails arcade shooters), not sure the RP3 is powerful enough for original Xbox games.

For driving, I am sure you’ve already been recommended the usual, but to reiterate: anything from Gran Turismo, Star Wars Episode 1 Racer, Collin McRae, WipeOut, Need for Speed, and Sega GT are all good picks.

IDK what games are actually compatible with your device, but I do have opinions about cool older games.

turn based:
Civilization
Master of Magic
Champions of Krynn, Death Knights of Krynn, and Dark Queen of Krynn
Shadowrun Returns, Shadowrun: Dragonfall, and Shadowrun: Hong Kong
Arcanum: Of Steamworks and Magic Obscura

platformers:
Noita
Dead Cells

driving:
Gran Turismo 4

I’ve been going through the old Final Fantasies. I finished FF5 and FF6 last year and recommend both. They have great pacing (They change things up before it gets too stale) and if you keep an online guide handy, it’s not as frustrating as it was when I was 10.

Alley cat

Lost Vikings is a classic. Go for DOS or SNES.

Driving:

Stunt Car Racer (Amiga)

WipeOut 2097/XL (PSX)

Both are top-tier racing games.

Milon’s secret castle. Top notch NES game.

Chrono Trigger is really a must if you like JRPGs.

Obligatory Frog theme 🐸

Colin McRae Rally 1 & 2 for driving games.

How are you liking the RP3? I don’t own a handheld myself but I’m always tempted to get one.

Breath of Fire was a great turn-based RPG series that I don’t think ever got enough recognition. 1-4 are all fairly solid (we don’t talk about Dragon Quarter), but 2 and especially 3 are my favorites.

Mega man 2, 3 (NES) and Megaman X (SNES).