Hey, welcome to the forums!
Nice 350 you got there! You might want to download (and print?) the service manual
if you haven't already. You'll need it to work on your Fourtrax, as they are quite a bit different than most other Honda models produced since. As for needed maintenance..? It depends on how well the machine was maintained by the previous owner.
Among the first things I might do:
Search the bike and replace any cheap china knockoff parts if any are found, with OEM Honda parts. China junk will leave you stranded early and often... and waste your time and money.
Check (and change) the gear oils in the front and rear differentials. The front diff must receive a gear oil containing a limited slip additive for the clutches.
Check and adjust the valves for proper (.003") clearance.
Check the wear on the camshaft chain.
Adjust the change clutch (1/4 turn) slack.
Check or replace the fuel filter.
Adjust every cable on the bike (per the manual) and verify that all vent hoses are in place, in good condition and all cables and hoses are routed properly.
Check the wheel bearings and swingarm bearings all the way around the bike. Clean & repack with grease and replace seals & bearings as necessary. Don't install any china bearings or you'll be replacing them again with quality parts, in a few weeks/months.
Check the brake shoes front and rear, adjust them and change the brake fluid.
Check the air pressure in each of the tires occasionally. The bike will ride and handle terribly if any one tire is low/high on pressure.
I can go on... but this might be enough for now. There are quite a few knowledgeable TRX350 fans here on the forums. Don't hesitate to ask for help or advice if you ever need any.
Thanks for sharing a photo of your pretty old gal!