Joe, welcome to OCC. This is the best place on the net for Cub Cadet info.
That is awesome that you got your dad's 782. I have my dad's 982 and it holds a special place in my heart....and the garage.
Your tractor was actually the 1318th built in 1985 so it was probably made in January. The engine was built in 1984. Here is a link to the engine manual if you need it.
http://www.kohlerengines.com.cn/onli.../tp_2043_a.pdf
The others have gave you some great advice. I would also pull the tunnel cover off and check the driveline for any wear and pull the rear cover off the rear and give it a good cleaning. That paint looks pretty good too and it should buff out to look showroom new.
I'm also liking that XT3 next to it. I have my eye on one if I can work the deal out. Keep feeding us the pics on your progress.