You may need a skinny-ended grease extension for some U joints. Also, a flexible, generally 90 degree grease gun head is more useful than a straight one unless you have a lot of fitting where you have to reach a lot with the straight one, then buy the flex or 90 degree extension.
You need to get a grease gun with a release on the tension spring that keeps pressure on the grease inside it, as the seal on the end of the spring will often leak grease past it if the spring leaves it in contact with the grease all the time. Maybe those battery powered ones folks recommended do this, too.
Just watch for leaks of transmission or hydraulic fluid. If no leaks, best not to ever check them because you can get dirt into the check holes during the checking. Similar for checking coolant. Don't if no leaks or don't need to if it has an expansion tank that you can see through.
I only check engine oil about once/week or after a heavy, long usage.
Check tractor for loose stuff, out of place, etc. every time you use it. Make sure that dip stick is turned so the handle is in towards the engine. I had brush remove the dipstick once and learned this trick.
Be sure you know your wire routing in case brush hits it and moves or removes it, so you'll know where to put it all back.
Make sure your radiator screens are cleaned regularly. Also, if you have a foam element over your air cleaner, remove it and wash out with soapy water regularly.
Ralph