Schwalbe blizzard or vittoria zaffiro 700 x 23c tyres will work well and dont break the bank. Try ebay and wiggle there is loads. Vittoria do them in yellow to match your paintjob. Giant inner tubes are only £2.50 each at wiggle.
The shimano HG50 chain is what you want, £10.99 from wiggle. Chain tools are £3 on ebay, very easy to do, follow this guide:
Try cleaning and loobing your front & rear mechs, and fit new gear shift cables, once you've got a new chain (and i would suggest fitting a shimano 7-speed Mf-tz21 freewheel/cassette (only £15 ish) then you'll probably find your gears work fine without renewing the mechs. They might need the mech adjuster screws adjusted at the moment, but you have to adjust them further to upgrade to 7-speed.
Poor gear operation is normally down to worn chain & sprockets, stretched cables, mis-aligned, dirty and badly adjusted deraillers and pulleys. If you replace them, the new ones would need setting up anyway
For shifters search for vintage friction downtube shifters on ebay. Avoid SIS shifters they probably wont work. They can be a pain to get to work and suss out which order the washers should go in.
I havent looked at your pics but you might get away with a thorough frame cleaning with white spirit, wet wipes, localised touch-up and/or spray paint & new decal set rather than a full respray (which is a bit of a PITA)
Another thing to consider is modern dual pivot brake calipers. The old single pivots are poor. Campagnolo veloce or centaur's are only £30 a pair, look the bizz, transform your braking and are a very easy worthwhile upgrade. If yours has steel/chromed rims the brakes will never work good however