For barrel cleaning I highly recommend the Air Venturi PatchWorm Multi-Caliber Pocket Gun-Cleaning Kit, Wet and Dry Patches with Ballistol. The kit has caliber specific pulls and comes with both wet and dry patches. Additional patches are available. Available through AMAZON & Pyrmayd Air

Be careful using petroleum or flammable lubes on pellets. They can cause dieseling and over pressuring possibly damaging seals or components.
Ballistol or silicon oil would be my first thoughts.

Seals, I use either silicon oil or a thicker silicon grease depending. If you use them just a thin coat works well. The grease will also help to keep seals in place during assembly.
Available through Amazon an automotive stores.
 
What are you guys using for silicone oil to assemble hand air pumps and also for lubing slugs and barrel cleaning as well? Thanks!
I use Super Lube Silicone Oil 100 cSt made by Syco Chemical Corporation, FWIW, for compressor and pump lubrication.

I have used some lubes for pellet lubrication but the results are inconclusive so no use to post them.