I've got a '62 Suburban that's my daily driver. i've also written for car magazines for 13 years, so I kinda know what works and what doesn't from an entry-level perspective.
You want to enjoy your truck. Mine still LOOKS like a basket case, but I'm based out of the Atlanta area and I've driven it to Joplin, MO, West Palm Beach FLA and Virginia, plus all points in between.
Absolute first thing you should do is replace the wiring harness. I speak from experience. A wiring harness will kill you. American Auto Wire has an EXCELLENT harness that is very, very easy to install. Anyone can do it, and it gives you GREAT peace of mine. ("I smell something burning?! Is that me?!!!")
Next, replace the rear gears. It'll help with that granny 4 speed. My '62 Suburban is my daily driver, so I went through all this. I have a later '81-87 4spd OD manual trans. It feeds a 3.42 Dana 44 rear end. The GMC Dana rears are bolt-ins for '60-'62 , and worth the hassle of finding one (if you own a Chevy). Dana 44 rear ends are still used in Jeeps, Explorers, etc. and you can find a 30-spline rear differential chunk just like I did on EBAY for pretty cheap. New ring and pinion gear, rebuild kit and someone to set it up, and you're on your way.
Next, I would do a power brake/dual reservoir master cylinder kit. Not glorious, but VERY worth the money and the effort. Dual reservoir master cylinders are well worth the cost.
after that, I would recommend a 3 row radiator, because over-heating while sitting in traffic sucks. follow it up with a 7-blade fan and a mechanical fan clutch, and hunt for a fan shroud.
That's the combo I have, and it does very well in heavy traffic.
After that, turn your attention to the engine (I have a 290hp crate engine), and then bodywork.
I drive my Sub every day, and absolutely LOVE it, even though it's a work in progress.
-Brad