I'd second what everyone was saying about ditching cotton. If you can get some merino that would be awesome. First Lite has a really lite long sleeve quarter zip aerowool that is great for the hot weather here. Most of my trips in Azone for deer, I wear Prana stretch zion pants or First Lite Corrugate guide pants, Darn Tough socks, merino boxers, and the First lite aerowool long sleeve. In the pack, is one short sleeve regular thickness merino shirt to add as a second layer, a puffy jacket, a pair of wool socks to stalk in (if bow hunting), and a merino beanie. When it gets colder, I will add a thicker merino long sleeve and long underwear to the pack. Lots of options for merino layers, most of my underlayer merino isn't camo and usually not a hunting brand. I am sure there are other lightweight merino like first lites aerowool, but I am not sure. All I know is that was huge when I got that, it was so nice to be able to wear a long sleeve all day when it was 100+ and it didn't stink like the synthetic stuff. I don't take extra underwear or socks. Usually for rain gear, I just take this cheap lightweight poncho, unless I know it is going to be storming for a big part of the trip.
I would definitely look into treating your clothes with Permethrin from Sawyer Products for the ticks and get a small thing of Picaridin bug spray as well
I would also suggest getting a set of gloves. I just use a set of gloves from the hardware store and wear them pretty much all the time while hunting. Great for pushing through poison oak or crawling through brush and oak leaves and hiking with trekking poles. Leading up to bow season, all my bow shooting is done while wearing those same gloves so that I am used to shooting with them. I do so much busting through brush and scrambling up steep rocky stuff where it is just good peace of mind not having to worry about your hands getting beat up. Sounds sissy, but it works, and I am a tile contractor so my hands aren't exactly soft, it's just one less thing for me to worry about and concentrate on hunting.
As for your gear, start with what you have and do a couple 1 or 2 night backpacking/hunting trips and start figuring out what you can live with that you have now, and what you need to change right away. If you are hunting Azone during the summer, you can probably ditch the fly on your tent for most trips to save some weight. Maybe get a poncho liner (woobie) from a surplus store to take for the summer trips instead of the sleeping bag. I was using a poncho liner the past 2 years and adding a costco down throw with it for the colder trips instead of my down sleeping bag to save some weight and it worked great. I ran that until I could get a nice down quilt which I just got and am stoked about. I ran some pretty shitty tents for awhile as well until I saved up some and caught a good sale at REI for a decent lightweight tent. Basically run what you have and slowly start changing stuff out, but definitely don't think you need all the greatest to start going.