Hi and welcome,
Oh geez! Hard questions to answer!
Lucky you! I'd love to "inherit" a GT6 (or pretty much any Triumph, not to mention more than a few other models from some other British marques!).
First of all, go out and get some books about car restoration in general. These can help you organize and get started. Then get books specific to the GT6 Mark III to help decide how you want the car to end up and educate yourself what to look for, how the car goes together and what constitutes "original", if that's important to you (an original car is generally more valuable). Next try to realistically decide what level you want to take the car to, set up a plan and decide on a $$$ budget and a time line (all of which *will* get blown outta the water anyway, but hey, go ahead and try to set some goals).
There are "restorations" everywhere from shade-tree, rolling, ongoing fixups of daily drivers to full-blown, frame off, concours level trailer queens that no one dares to drive. What you do is up to you (and your budget).
Some critical things are adequate space (Until recently, my TR4 frame-off projekt pretty much occupied all of a two car garage... a three car garage would have been ideal!) and a lot of organization! Photograph everything before disassembling. Make sketches. Take notes! Get a good manual specific to the car. Clearly label parts as they are removed, bag them and store carefully. A bunch of deep storage shelves would be handy, too. Throw nothing away until the job is complete. Even obviously worn out and unusable items will be needed for close comparison with replacement parts.
Now, don't watch "Overhaulin" or "American Hot Rod" on TV. These shows will make you think the job is easy and can be done in a week (in the case of the former) to a couple months (in the case of the latter). It can't, unless you have major $$$s to spend! Note that Overhaulin has 30-40 people working on the car round the clock for 7 days. They also job out a lot of the work. Do the math and you'll get some idea how much time these pros put in on a restoration project, then double or triple it for an amateur. By the way, note that many of the pros are highly specialized, while most amateurs try to "do it all".
The engine can be "boiled" to clean it out. This will necessitate installing cam bearings, but a good machine shop can handle it. Good opportunity to upgrade some things in the engine, if you wish.
By all means, rebuild the engine yourself if you wish. Triumphs are a good learning ground. They are relatively straight-forward and comparitively easy to work on. It would be useful to have someone experienced looking over your shoulder (figuratively, if not literally) as you don't want to have to do the job over, if at all possible, due to some silly mistake or oversight! Having an engine professionally rebuilt is pricier, but generally more certain and often warranted to some degree. You will become good friends with a local machine shop and get to know some parts suppliers, along the way.
An interesting possibility would be to take some automotive classes at a local junior college, and drag in projects from your car to work on. These can be great ways to get help and supervision of your work, while still doing a lot of the work yourself. Useful classes might be in engine repair, body work, painting, welding, sheetmetal fabrication, etc., etc.
From the sound of it, I think you are likely looking at replacing or repairing the frame. If it were me, I'd start by looking for a good straight candidate to replace it with. Once you have that, send it to a frame shop to be professionally checked for trueness and soundness, do or have done any work on it that's needed and then have it painted. After that you can start to remove parts from the old frame, install them onto the refurbished "new" frame a little at a time as each item is rebuilt/restored/replaced as needed. A frame off job like this is not so intimidating as it might sound... really goes pretty quickly and gives a method of organizing... but does require a lot of floor space! Try to visualize how the factory built the car, and that might give you some idea of the process.
How much will you spend? About 3X your wildest estimate!
Seriously, it totally depends on how you go about the restoration and your end goals. I've seen amazing cars put together for under $10K, but that involved a lot of luck and skill. At the other end, the TR4 recently sold for $97K at Barrett-Jackson auction *supposedly* had $85K spent on the restoration. I'm still trying to figure out how someone could spend that much! A more common figure for later model Triumphs might be in the $15K-$50K range. That's a big range, I know, but that's about as accurate as is possible without *a lot* more specifics.
Nearly always, selling the car won't recoup all the cost of restoration. Completed cars that are "original" or only have very high quality modifications will bring the best returns. Only very high-profile professional restoration shops that can both do the job as efficiently as possible *and* charge top dollar ($60K, $80K, $100K and up is not uncommon) for their work can really be profitable in this business.
The good news is that Triumphs (and many collectible cars) are appreciating. The older cars and rarer models will do so faster. GT6 are relatively rare, compared to say TR6, Spitfire and TR7. But, they aren't as "retro" as a TR2, 3 or even a 4.
One nice thing about Triumphs overall is parts availability. There are a number of suppliers of both new and used parts. (It pays to shop around.)
What's a restored GT6 Mk III worth? I dunno... check Hemmings to see what people are asking for them, but that doesn't give a clue as to the actual selling prices. Check eBay for recently completed sales, but seller descriptions vary wildly: it seems there that one seller's "concours" or "NOS" is another person's daily driver or used and rusty part. Auction prices can be very fickle, anyway. I paid just over $400 for an old camera that usually sells for $2000-3000... simply because the seller's description and headline were so poor that no one saw it and I had little competition. On the other hand, there are instances where people go crazy bidding against each other (see the $97K TR4).
I can assure you, though, likely the cheapest would be an abandoned or incomplete restoration. Way too many well-intended projects end up that way, and are sold for pennies on the dollar.
There are old car Blue Books that might give a better feel for average prices.
www.kbb.com might help, as well as Hemmings and Edmunds (not sure I spelled that last one right).
You gotta really love doing the work to make it to the finish line of a major restoration project.
Depending upon how familiar you are with the car already, maybe you'd be wise to join a local Triumph club or otherwise try to get a ride in, maybe drive some GT6s to see if you really like them and want to put in the time and expense. For example, if you are 6'-5" tall and have a 40" waist, it's likely not the car for you! (And you should probably just ship it to me /ubbthreads/images/graemlins/wink.gif)
Car clubs can also be a great source of advice and assistance - not to mention encouragement - all of which you will need along the way. This website is also a great bunch of people with tons of knowledge about these great old cars! There are many more resources on the Internet, which has really been a boon for old car lovers and restorers! (That I can assure you from trying to get Triumph parts back in the 1970s.) There are also a lot of *opinions* (mine for example) here and elsewhere. This can be everything from do/don't powdercoat the frame, to do/don't use syntheic oil, to do/don't use this particular camshaft. More info, but more to wade through and sort out.
Oh and one other thing you might want is a BMIHT "build certificate" for your car, available at
https://www.heritage-motor-centre.co.uk/archive/certificate_index.html Based upon your car's VIN number, it lists nearly all the original specifications as the car came from the factory: colors of interior and extrerior, options installed, extras that were included, along with shipping and destination information. Sometimes even key codes are given. These documents can be very useful if you want to build the car to original specifications.
Good luck with the project. We'll be wanting to see pictures and hear regular reports, you know. /ubbthreads/images/graemlins/cool.gif
/ubbthreads/images/graemlins/cheers.gif
Hi Guest!
Hey - did you know if you click on the title of a thread it will take you to the first unread post since you last visited that thread?
but were afraid to ask: 
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smilie in place of the real @
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