I wear a pair of cheep leather gloves. I wipe off the grease fittings with them and what ever else needs wiping. Once they get too dirty I rub a healthy dose of hand cleaner into them and throw the gloves into the wash with my work clothes. I let them air dry and reuse them till they are worn out.
This keeps my hands and clothes relatively clean enough so that they not usually an issue to operate equipment.
The grease gun nozzle usually gets wiped off on the grass or tire or what ever else is handy. If it's really bad then a wipe with a paper cloth is usually sufficient to clean it up. Once in a long while the whole grease gun will get a bath in my varsol tank or more often lately just a good spray down with brake clean.
I'm of the opinion that a lot of people think that if a little grease is good then a lot is better. My mower deck bearings only get 2-3 shots. Any more and you run the risk of pushing the seals out. Any needle bearings don't need more than the same. Bushings I like to see the grease come out somewhere but I'm probably over greasing them. U-joints get grease till it comes out all 4 of the needle bearings. That way you know that all 4 got grease.
I don't grease the telescoping part of u-joints at all. [[[Spraying WD40 is good enough for me.]]] I hate it when the telescoping part gets so full of grease and dirt that it's a struggle to get it to move. Yes it wears out faster the way I'm doing it but it won't wear out in my lifetime. I'm good with that. I lube my pto connections the same way for the same reasons.
If you ever pressure wash your equipment make sure you grease it and use it a bit after. It pushes the water out of all the bearing surfaces. Your ball bearings will thank you by lasting longer.