I've planted mine already. I usually wait till Oct. 10th or so, but if I don't get it done now , it won't get done cause we are going on a fishing trip this week then a business trip soon after that.
I think any straw will work for mulch.

You'll be planting soon,I'd think.Here in 6,I like to start mid to late Nov.,but have gone into mid Dec.
Of all the mulches used on garlic,I like chopped leaves the best,about 4" at least.I'm getting scapes by the time any notable weeds develop.

I usually plant around Columbus Day so anytime in mid October should work well. I also like using shredded leaves ... I try to cut the grass short a week or so before heavy leaf fall and then just run the lawnmower over the grass and leaves and catch them in the bag. Mostly, this reduces the amount of any weed seeds in the grass mulch and I use about 4-6 inches of the grass clippings over the garlic planting. By spring, it's down to around 1/2 inch and the clippings suppress weeds for a while. I'd probably remove the mulch in spring if you have slow draining soil since it does hold a bit of water.

I can't do straw mulch - every time I've tried, I get mice taking up residence.

I use pine shavings - easy to obtain and a bale pretty well covers my planting. Nice and clean - even smells good!
Columbus Day is my target for planting.

Do you completely bury the garlic bulbs and cover them with mulch? or wait till the shoots are up?

I plant my bulbs about 2 in. deep and then cover with about 6 in. of mulch. I pull most of the mulch back in the spring, but leave enough to help suppress weeds. Keep in mind that I am in zone 4.