Definitely Autobleem to start with, I'd say. Run it from a USB 2.0 stick in the right hand controller port.
But.. to enable firmware changes you actually use the Bleemsync image.. this backs up your firmware etc..
I'm using a firmware mod to enable me to plug USB 3.0 256Gb stick into a y-cable at the rear of the PS1 Classic. This means both controller ports are free for 2 players - but you don't need to do this initially.
For AutoBleem, use hi-res mode and scanlines on 75% brightness for the best look on most 3D games. You may find the PSC isn't up to handling a few games on this display mode and will frame drop occasionally.
In that case set it back to standard resolution mode and disable scanlines.
Most pure 2D titles like Metal Slug, In The Hunt, Mr. Driller, the arcade museum / collection packs will also exhibit a few tile edge lines with hi-res mode on, so they'll need to be run in low res.
I've not tried wireless controllers yet, but I'm told they work fine with a suitable USB receiver.