Outers Nitro Solvent is what I typically use to loosen carbon deposits in the bore. I wet the bore with it before I use a bore brush.
I haven't seen much difference in nitro solvents. They all seem to operate at about the same level of effectiveness. The only one I could tell was really better than the rest was the old BreakFree Solvent. I can't find it anymore. It was perfectly clear--looked like water, had an unusual sweet odor, and came in a small glass jar. I still have a small amount somewhere but they must have quit making it.
I use Hoppes Elite or M-Pro 7 for removing copper. I can't tell much difference in how they work. BreakFree or Outers foaming bore cleaners also work well for this. I generally alternate a 10-15 minute soak with one of these four cleaners with the Nitro Solvent and some bore brushing. These aren't particularly aggressive copper removers, but they do a good enough job that I rarely use anything else for copper these days. A nice side effect is that these cleaners have a very mild odor and are non-toxic. These cleaners have dramatically reduced my bore cleaning time and effort. I do almost no bore-brushing now compared to what I used to do.
I have tried Hoppes Elite Foaming cleaner and it's a bust, IMO. I had high hopes after seeing how effective the regular Hoppes Elite cleaner was but for some reason they changed something in the formula and the result is pretty useless.
For really stubborn deposits, I use RemClean. You MUST follow the directions on this stuff and shake the bottle well before using. It contains a mild, non-embedding abrasive that has to be shaken into suspension before you put some on a patch or it won't do much of anything. Put this stuff on a tight-fitting patch and work it back and forth in the bore and it will remove the final stubborn remains of metal fouling or carbon deposits. I don't use it on a regular basis, just when I can tell that there is fouling that's not coming out with anything else. Be sure to flush any remaining residue of the cleaner out of the bore and try not to get it into areas with moving parts unless you can be sure you can completely clean it out afterwards.
I have cleaned with only BreakFree CLP in a pinch. It's a good nitro/carbon solvent but does nothing to remove copper fouling. It's what's in my range bag.
For general cleanup (wiping off fouling) I use different things at different times and for no particular reason. Hoppes Elite or MPro 7 on a paper towel or an old rag works well. You will have to wipe the cleaned parts down with an oily cloth when you're done as both of these cleaners will leave the metal unprotected.
Ballistol or BreakFree CLP work well for wipedown/cleanup if you want something that's going to leave a residue of lubricant when you're done. Ballistol doesn't seem to be quite as good a cleaner and from what I've seen in testing it's not quite as good at corrosion protection as BreakFree CLP. I've been using it lately because I bought some to try it out and haven't used it all up yet. I usually use BreakFree CLP to wipe down all metal parts when they're clean to leave a very light oily residue for protection against corrosion.
I have a number of lubricants I use on handguns. Dillon Snake Oil is a good oil that doesn't evaporate and has virtually no smell. It's also non-toxic. Lubriplate FMO350AW is another oil in the same vein which probably offers a little better corrosion protection. BreakFree CLP does just fine for this application as well.
I also use light grease for areas where friction is obviously causing wear marks or where it's obvious that there is the potential for significant wear due to metal-to-metal contact. There are a lot of different light greases out there and I'm not sure that there's a huge amount of difference in how well they work.
A light white lithium grease like Permatex sells in the small squeeze tubes will work well. Lubriplate makes a grease called SFL-B that is very good and is also non-toxic. Both of these greases have the advantage of not being too bad about staining materials they come in contact with. Tetra gun grease is ok as well but I didn't care for it because it seemed a little less "durable" when I used it.
For most handgun applications, a good anti-seize will work. Just be aware that they can be really messy to clean up and seem to get on EVERYTHING that comes in contact with the object being lubricated.
Brownell's Action Lube and the other moly-based/moly-containing greases work well but are messy like the anti-seize greases are.
I generally shy away from greases/lubes that contain graphite because some sources indicate that under some circumstances it can interact badly with aluminum. Check the ingredients because it's not uncommon for grease/anti-seize products to contain graphite.