Whenever I acquire a piece of equipment I go through it top to bottom. I always start with a full service.
Wash/ Degrease everything
Engine Oil + Filter
Hydro Oil + Filter
Fuel (Drain old fuel and replace with new) + Filter
Clean/ rebuild carburetors as necessary
Check belts and replace as necessary ( I always have extra belts on hand just in case)
Sharpen/ Replace Blades
Check tire condition and pressure.
Then I replace/ rebuild anything that is worn or looks like it may cause a problem.
Thats just a basic list off the top of my head. I like to be pretty through when I go through everything. I would rather deal with potential problems not then have equipment go down during the season.