Before you ride it, pull in the clutch and kick it a few times; after the first kick or two, the starter pedal should "slip through" without engaging the engine. Classic Triumphs can get you in trouble if you don't check this, ESPECIALLY after they've sat.
If all is right with the bike (clean gas, clean carbs, good battery or 2MC capacitor, clean points, and oil tank 3/4 full), the bike should fire in a few kicks (bonus points if it starts on the first kick).
Otherwise,
Flush out the gas tank and overhaul the carbs with new gaskets and viton-tipped float needles. I use the one gallon can of Berryman's Chem Dip, set it in the sun to warm it up, then soak carb parts for 30 minutes. Flush thoroughly with running water. Use fresh mid-grade fuel. Non-ethanol if available. Lube the throttle cables; throttle should snap back when you release it.
Get a BIG tin turkey pan and set it under the engine; pull the sump plug underneath. It's a big nut, angled; USE THE CORRECT SOCKET. Inspect the screen closely before washing it, see if there are any metallic particles, then test with a magnet and see if any are steel. Open the valve lash inspection caps and dump some clean oil in until it flows out the bottom relatively clean. Replace the sump plug.
Drain the oil tank, remove the large screen fitting and check for debris. THOROUGHLY flush the tank with 50/50 Simple Green & water. Look in with a flashlight and make sure it's clean. Tie a small rag to some sort of long instrument and use it to swab out any remaining sludge & residue. Replace everything and re-fill with 2 quarts 10W40 Castrol 4T oil.
Open up the primary side engine cover and drain whatever is in there (likely milky sludge). Note the position of the 3 clutch center screws and remove them, pull out all the plates using a very thin hooked dental tool and/or magnet. Wash the plates in 50/50 Simple Green and water, place them in the sun to dry. Re-install them dry, screw the clutch screws back in to where they were. Adjust the primary chain to just snug, replace the primary cover. Pour in 1/4 cup 10W30.
Check and adjust valve lash by the book. If you don't have a compression tester, you can borrow one from AutoZone. Run a simple cold compression check with both spark plugs removed and the throttle held wide open. As long as both cylinders are above 100PSI and relatively close to each other (10%), you'll be okay as a starting point.
There is a "double nut" plug on the transmission (underneath); pull it and drain the tranny. It will not smell good. Hopefully, what drains out will resemble heavy oil, but it is likely to look more like dark brown mustard. remove the small nut from the drain plug and re-insert it, then slowly pour some Castrol Hypoy oil in until it starts to drool out the small plug nut. Replace the plug nut.
Carefully clean the points, don't even try to adjust them yet, as the bike may run without fiddling with them. It probably doesn't need a battery as long as the large blue 2MC capacitor hasn't burst or bulged excessively. Pull the old spark plugs out and buy new Champion N3C (802) plugs. Check for spark after cleaning the points, I'll bet it will spark after the 2nd or 3rd kick to charge the 2MC capacitor.
[ONLY FOR STANDARD FORKS: Flush the forks by unscrewing the drain screws, holding the front brake lever, and pumping up and down to dispell whatever is in them. Most likely it will be milky sludge, but you might get lucky! pour in 1/4 cup of clean 10W30 in each leg and pump some more to flush them till they drain reasonably clean; then, refill per spec.]
IF the tires are at all sound, air them up to 30 PSI; otherwise, install new tire (I recommend pattern Dunlops, they are well made and very affordable, and look exactly like the originals).
Clean the chain and lube it with good sticky lube, then wipe down the excess.
Adjust front and rear brakes to provide good braking force with as near zero rolling resistance as possible (a fine line, to be sure).
You MIGHT be able to fire the bike up and take it for a spin now!
No, I didn't just type all that, I copied and pasted it from another forum where I just gave the same advice to another new guy...