Newbie tuning and repairing my Viscount

Hi Dan, welcome to the fun! I looked through your photos, that's a nice bike, if a bit rusty. Depending on what you want, I think you have two basic directions -

1) - The cranks look to be sitting on an original, press fit, sealed bearing bottom bracket and axle. it spins freely, so leave that alone until the cartridge bearings go bad.
-the wheels look to be Viscount, front and rear. The spokes are rusty. As long as they spin freely, I'd use them until the spokes start breaking, then rebuild that rear wheel. In the meantime you can rub down the spokes and both quick release skewers with some rust remover and the rough side of an old kitchen sponge.
-it doesn't look as if the wheels need much, if any truing. Lucky you!
-replace the cables and housings as needed to get smooth operation. Brake cables and housings, especially, have come a long way since the 70's.
-the shifters may need to be disassembled and cleaned, as old oil and grease can cause sticky problems as you describe.
-replace the foam with some current tape - your shop will have choices, and the various brands will be similar. Planet Bike is, at least around here (Chicago) relatively inexpensive, and it's easy to clean if you like colors other than black, but it gets a little slippery in the rain. The Cinelli and it's copies that use cork soak up a bit of water, but I think they stay grippy. That's what I prefer to use on high mileage / bad weather bikes.
-if the tires are dry rotted, replace them. Otherwise ride them until they wear out or start giving you too much trouble.
-consider Kool Stop brake pads
-have the shop look at and measure the chain for stretch. Clean and lube, or replace, as needed.
-clean and lube the freewheel
Ride Ride Ride

That's the quick and easy path to a nicely rideable bike.

2) If you want to turn it into a higher performance show stopper -
-have a good long talk with the folks at your local bike shop, and compare their recommendations to these -
-since it's from the 70's I'd keep it rather completely original and period correct
-consider new brake releases to replace the rusty ones, eBay or Amazon might be good sources.
-consider (as I learned on this forum) Velo Orange alloy quick release wheel skewers to replace the rusty Viscount ones and save a few grams (but lose the distinctive "V"s, which you may want to keep.)
-or see how to best remove as much rust a possible from them and keep going.
-cables and housings, of course
-Cinelli cork tape
-Panasonic Pasela or similar 27" tires
-Kool Stop brake pads
-consider rebuilding the rear wheel now, and the front later, with stainless spokes and newer rims. This is usually not cheap, so I'd wait until I need new rims before I rebuild the wheels, but it's your decision. I'd stay with 27" rims as they will switch in and out with no changes needed to your brakes - Sun CR18s are particularly nice and look to be quite durable, too.
-consider a new chain and freewheel
-consider replacing the SunTour rear derailleur with a Shimano Crane which would match the front changer and the shifters. Personally I much prefer the SunTour, so much that I'd replace everything else and leave the VGT, but that's me.

Please keep us informed as to what you decide, and photos of your progress or at least the finished bike. 

Best wishes,
Frank

That's a good looking bike - the bar end gear shifters are a nice bonus.

Just a brief observation from my experience - be a bit choosy if you're buying a new rear wheel and hub. The spacing between the rear dropouts on my Sebring is (was?) about 124mm. I bought a new replacement 27" rear wheel (new hub, alloy rim) from a reputable cycle shop and fitted it to replace the original chrome (rusted) steel rimmed wheel on my Sebring. Unknown to me, the new hub was about 128mm from nut face to nut face. I noticed that I had to 'spring' the rear triangle a bit during fitting, but thought nothing more of it until I noticed that my rear tyre was rubbing on the drive side chain stay. Thinking the QR had slipped, I took a closer look and found that the drive side chain stay had broken as it meets (met?) the rear dropout. I've yet to find a cost-effective way to repair the broken chainstay, so my Sebring frame is now 'wall art' in my shed, I'm sad to say. Lesson: check the hub dimensions before buying and fitting a replacement wheel. Also see Sheldon Brown advice, or try learning how to rebuild a wheel on your old hubs with new rims.

On a literally brighter note, I have just discovered the magical powers of aluminium ('tin') kitchen foil for cleaning up tarnished and lightly-corroded chromed forks & other parts. Just scrunch a piece of foil gently and rub the chrome with it as if you were polishing it. This has worked a treat on my chromed steel forks and brought them back to lustrous life. It's certainly worth a try on your rear QR lever. It also works on aluminium, apparently.

Best wishes to you, and another vote from me for the restoration approach if you can find a sympathetic bike shop or internet parts supplier. Some local shops will take one look in their current Shimano catalogue and tell you that the part you want isn't available any more - don't accept that, as it will be somewhere, and this forum can help you find it. It can be very satisfying when you pass the carbon-fibre riders on your 1970s classic (then turn off their route quickly to get your breath back!).

Welcome to the forum Daniel.
I don't think there is a setting to get email notification, but I could be wrong.
I'm on my phone so can't really address some of your points but will certainly reply when I eventually get home.

Update - there is an email, but generally quite a while after the posting.