The AHA has instructions and video on how to make a cooler mash tun. They use a rectangular cooler but the process should be similar to round coolers as well. This would be one the simplest/ cheapest routes for a "traditional" batch sparge cooler mash tun.
https://www.homebrewersassociation.org/pimp-my-system/building-a-mash-tun-has-never-been-so-easy/
If you are looking for something fancier but aren't great at DIY solutions, you can purchase ready made bulkheads to create threaded fittings onto the cooler. Then you would only need to add a valve on the outside and something to filter on the inside (threaded barb and hose braid, manifold, false bottom, or BIAB bag).
Parts list for this option would include:
Bulkhead
Valve
1/2" NPT x 3/8" barb fitting
6+" stainless water supply line (toliet, dishwasher, sink water line).
Ready made bulkheads can be found at your Local homebrew store or online at
brewhardware.com,
bargainfittings.com, or many others.
You can build your own bulkhead from local hardware store but most likely you'll find only brass fittings. Brass works just fine (It's what I use in mine), it's just not as pretty.
As far as fermenting equipment, you'll need:
food-grade bucket (Bigbox store or preferably LHBS)
bucket lid
gromet for the bucket lid
airlock
siphon
food grade vinyl tubing that will fit onto the siphon
morebeer.com,
austinhomebrew.com, and
love2brew.com have solid equipment starter kits. They'll have most everything except a kettle, mashtun, burner, and bottles. Bottles can be had for free by collecting empty craft beer bottles.
To manage fermenting temps placing the bucket in a large tote of water will help smooth out temps. You can swap out frozen bottle to get the temp down lower.