Dave,
Welcome. First off, why don't you tell us a bit about your knowlege and experience with Healeys in particular and British sports cars in general. The n we know how much detail to give you.
If you really are new to these cars, you need a little attitude orientation. These cars are not Hondas or Toyotas or even Chevys. They come from a time when regular owner maintenance was expected. If you follow the maintenance schedule and do regular inspections, the cars run fine. If you treat it like a modern car that comes with a 10,000 mile oil change schedule, you'll have an unsatisfactory experience. The good news is the cars are relatively easy to work on, and people here are happy to give advice. First piece of advice: get a workshop manual and an owner's handbook. Both are available on lione through ebay sellers or through the catlogue companies.
The big Healey has a robust engine. They are legendary for sitting for years and then firing right up with very little effort. But you want to be safe and you want to minimize the impacet of starting a "dry" engine. First, drain out the old gas. There is a drain bolt at the bottom of the fuel tank, but if the area looks questionable (rusty) you can end up rupturing the tank by trying to remove a stuck drain bolt, so be careful here. Another method is to siphon the gas out of the tank. In any even, add fresh fuel slowly, all the while checking for leaks at the tank.
With a good battery installed (check to see if it is positive or negative ground before your remove the old battery and put the new on in in the same configuration) turn on the key to the first stop (don't crank the engine) and listen for the fuel pump. It makes a ticking sound. The fuel pump should tick for a few seconds and then stop. If it does not stop, the fuel system is leaking gas. This is a common problem with Healeys that have been sitting. It usually means the float needles are stuck open. Look in the manual to find out how to unstick them. It could also be a leak in the lines, so find the source of the leak and then address it. Once the fuel system is found to be sound and safe, move on to step two.
Remove all the spark plugs (buy new ones. May as well replace them) drain the old oil and change the old filter. Refill the engine with fresh oil and spray some light oil into each spark plug hole. Now disconnect the distributor wire. In this configuration, crank the engine for about two or three minutes while someone watches the oil pressure gauge. Keep cranking until the starter motor has turned the engine enough to build up oil pressure. If the gauge never moves, it could be brocken. Disconnet the thin oil pressure gauge line coming from the engine at the engine and then crank it again. If oil shoots our of that lime, you have oil pressure, but a problem with the gauge.
Now the engine is lubricated and the start-up won't damage the bearings, etc. Replace the plugs, reconnect the wire to the distributor and remove the distributor cap. crank the engine over while someone looks at the distributor and see if you have spark at the points. You should replace the points, rotor condensor and cap, but for now, see if you have spark. If you do, replace the cap. Now drain the radiator and the engine block and flush the coolant out. They both have petcocks that let you drain the coolant. Refill with fresh antifreeze mix.
Now put the car in neutral, set the chokes, turn on the key, and start the car. If it runs poorly, then you need to check the ignition, compression and carbs to find out why. If it seems to run well, try operating the clutch. See the next paragraph.
The hydraulics are another story. At the very least, you will need to flush the brake and clutch hydraulics and replace the fluid with Castrol LMA and bleed the system. At the worst, you will find lots of leaks and have to rebuild all the hydraulics and replace the lines. Usually, you end up needing ro replace some of the rubber lines, and you should rebuild all the cylinders and calipers. Your saftey is on the line here, so don't skimp on this step. With the hydraulics working, see if the clutch works. Often the clutch disc sticks to the flywheel after sitting for a long time. You will know if this has happened if the clutch pedal operates, but the car will not go into gear. You will hear a grind. If that happens, the disc can be unstuck, but I'd seek help from soneone with experience.
You should also drain and replace the tranny and diff oil while you are at it. I think that vcovers most of the basics for bringing an old sleeper back. Oh, you should also replace the tires before driving the thing around much.