Welcome to LRH! I currently use a foaming Bore Cleaner/Scrubber type first. Hoppe's and Birchwood Casey both make one and there's probably others. I let this set for 15-20 minutes, use a dry patch to push most of the sludge out the muzzle (put a rag on the floor or use a catch bottle).
Note on patches/brushes; I only use nylon brushes, no metal types, and it's two sizes smaller than the caliber of the cartridge I'm cleaning (i.e., .270 Win gets a .25 cal brush).
I feel this step gets 90% of the carbon and copper out of "my" barrel(s), maybe more.
Then I use a wet patch with an ammonia based copper remover (Montana Extreme, household ammonia) immediately following up with enough dry patches until it's apparent it's gone.
At this point, (almost done!) I use a carbon remover, especially so on the over-borish cartridges (6.5-284 & .270 Allen Magnum) to remove and further prevent a carbon ring ahead of the chamber. Of all the steps I have listed this is the most intense, effort wise. This has been an issue with my 6.5-284 Norma, even though I regularly cleaned it of powder residue and copper after < 50 rounds.
I have used Sea Foam a lot and it seems to work pretty darn good. I have recently found and started using Break Free CAC (Carbon Cutter). Jury's still out. 2-3 dry patches afterwards. Even though this removes the majority of these two products, I use 2-5 fouling shot's to get the oily residues out. Most times the first 2-3 shots will be higher than the established zero/POI.
So there is my cleaning routine. Use it as your own or cut/paste as you like. You'll probably find that the routines vary as much as the barrels and shooters do. Happy shooting.