Though this is a big project, you could probably felt it by hand if you wanted to. I used a front-loading washing machine, which takes a little longer and often doesn't felt as thoroughly as a top-loading machine.
Whichever way you choose to felt, you get to decide how far you want to take it. On my bag there's still a decent amount of stitch definition but the bag is firm enough I wouldn't worry about stuff poking through.
If you want a bag with much less stitch definition, know that it will come out smaller.
Once your bag is felted and dried, take your cotton trim and weave it in and out of the eyelets and trim to length, leaving extra length for tying the bag shut.
Attach your bag handle however and wherever you would like. Mine is one of a pair of bag handles and it's simply strung through the trim in the place I wanted the handle to fall. (You can also sew it in, of course.)