Welcome to the wonderful world of vintage electronics. As jpdylon already pointed out, you don't need to be an electrical engineer to fix up and maintain these vintage electonics, but a basic background in electronics is a big plus. Even if your stuff works at the moment, it will work so much better once the faulty/flakey old capacitors have been replaced. These old things can have a very nice picture and great sound once repaired. To start things off, I would reccomend going to
http://www.antiqueradio.org/begin.htm and reading some of the articles, most especially the one about replacing capacitors. If you don't know electronics now, then start reading, you'll find it can be remarkably simple to learn.
Phil's site mainly deals with radios, but he has some TV's as well, read his restoration logs - they're great and are full of useful information. Old TV's and radios are pretty similar from a component standpoint, but radios are much simpler. Start off with a typical 5 tube AM radio set first, and work your way up.
You don't need fancy tools to fix old radios and televisions. Some of the fancy equipment like a tube tester or a signal generator are really nice to have, but far from necessary. You do need some basic tools - the essentials are :
Mulitmeter (cheap Radio Shack model is fine for most things)
Screwdrivers (flat, phillips, couple different sizes, get decent quality tools, I love Sears Craftsmen tools)
Nut drivers, especially 1/4"
Pliers, wire cutters
Soldering iron - I know some people like expensive soldering stations, but I have been using the 30 watt Radio Shack iron for years and it works great for all this stuff with practice. The tips are short lived, but cheap. The iron kinda gets crappy after a year or so of heavy use, but they're only $8.
Solder Sucker - Radio Shack has these, they're about $8, and are a little pump thing to suck up exra solder. Some people never get the hang of it, but I love it. Great tool. Once again, they break after a couple years, but they're cheap and work great until the screw threads holding the top part into the bottom strip out.
Solder - get 60/40 rosin core - make sure it's real solder, not that crap lead free junk, and NEVER use acid core plumbers solder.
Coffee, for late night/early morning/afternoon/mid-day tinkering.
A trip to Radio Shack and Sears can get you all the essential tools for under $60.
Get the schematic for your radio/TV/etc. It is really nice to have when working. Most TV's are covered in Sam's PhotoFacts, but older models can be found in Riders. Your Dumont TV is in Riders volume 10. It's on NostalgiaAir - here's a link to a PDF file of it -
http://www.nostalgiaair.org/PagesByM...7/M0004387.pdf
And, there's plenty of resources out there on the net, as well as here. It's a great hobby, with a lot of fun things to learn about. It's pretty easy to get started, and we're here to help. What's your electronic background? Do you know how to solder or read a schematic? If you don't, then you'll have a bit of learning to do before you can get started, but don't worry, it's not too hard, and once you learn how this stuff works, a lot of other stuff makes sense too.
Have fun!
Also, post pictures of your Dumont. We love to see pictures
-Ian