Welcome to Permies Chris!
Like Mike said, I leave the roots in the ground for extra nutrients for the soil! It will just get better with age.
I cut down the plants at the base with pruners and depending on their size I will either just leave it there on top as mulch or stuff it into my compost bin. I'm in the suburbs right now so I use a metal trash bin for my compost.
When you replant, depending on what your mulch material is and what you are planting you may or may not have to move the mulch aside.
For my own garden I pretty much exclusively mulch with
wood chips since I can get them free so I have to push it aside in order to plant new seeds. Saplings are not strong
enough to push aside a heavy wood chip piece. If you have
straw or some other light material as mulch you may be able to directly plant certain seeds that have strong saplings like beans. But it all depends.
And of
course it is different for transplants! Push the mulch aside, transplant the plant, and put the mulch back within reason. I'll try to put the thinner more decomposed mulch that you
should be able to find underneath the top layer by now directly around the stems.
My biggest challenge here is being in hot hot Texas and wanting to push the mulch aside for direct seeding which then leaves the soil to get dry way too fast and I've become spoiled from the mulch and will forget to
water those areas. Oops.
I'm not the most experienced but those are conclusions I have come to after my two years of vegetable
gardening :) (Yes, Covid got me into
permaculture, yay!)