Although some RVers would never put anything but Michelins but the price premium was higher than I wanted to pay. I put Uniroyals on mine and, according to "my tire guy" are a good alternative to Michelins, albeit with somewhat less tread life expectancy. For most of us, our tires will age out before they wear out.
Toyos, Uniroyals, Goodyears or any name brand tires will be just fine. The same goes for shock absorbers. You'll read a lot about the merits of various premium brands like Bilstein and others but, IMHO, the OEM shocks are just fine. You'll read a lot about suspension upgrades, swaybars, etc. but leave that for later. My 2002 Itasca Suncruiser 35U (same as Adventurer 35U) does just fine without them.
In addition to the things mentioned by the other respondents, your highest priority will be to find and take care of any leaks. Re-caulking all your roof penetrations, vents, skylights, etc. will need to be done ASAP. Read up on the issue of sealing the roof edges where the fiberglass fits into the aluminum extrusion. Winnebagos are notorious for windshield leaks, which are another thing that needs to be taken care of immediately. There are numerous threads on all of these issues on Winnieowners.
Jim_HiTek has a blog with a lot of good info on some of the above and other topics:
RV Section - Repairs & Optional | Chaos Leaves Town
(Roof-related repairs are listed under section B: Chassis Repairs & Fixes)
Also, you'll find links to a lot of good info and resources in the "stickies" here:
If you don't have one, you can find your Operator Manual here:
And, one more thing, go to the User CP and create a signature that contains info on the year and model of your MH like mine and those of
a couple of the other respondents. This will help people respond to your questions. Adventurers come in many flavors.
Good luck and we look forward to more from you.