re: "Any pointers on simply driving and enjoying the car?"
In a way, you kinda answered your own question: Drive the car--no matter where and for how long--and you will enjoy it. I recently sold my late father's '65 Mustang convertible and '55 T-Bird and, although they were great cars they weren't the same experience. Both felt like yachts compared to my Healeys, a '56 BN2 and a '67 BJ8, and the Healeys easily 'out-handled' both of them (and, although I only got a brief ride as a passenger I felt the Healeys were much more nimble than our neighbor's original Mercedes Gull Wing roadster). The BN2 is more the pure sports roadster, and the BJ8 is more of a grand tourer, both fun in their own ways but the 6-cyl car is best on long trips for its 'legs.' Until a couple years ago me and my BFF took a long Healey trip in late spring; the latest was over 5,000 miles in about two weeks. We only put the hood (convertible top) up in the most severe weather, expected to last over an hour; we found that if we could keep speed up to 60 or more we wouldn't get terribly wet, at least after I rectified some leaks (the back 'seats' however, were a different story). It's the next best thing to being on a motorcycle, and I'd argue better because you don't have to wear a helmet, you may well be 'closer to the road' and can haul (slightly) more stuff and are arguably 'safer.' We stuffed the boot with spares and tools, and got by with a couple duffel bags in the rear seats.
Despite the 'horror stories' you may have heard about British cars, Healeys are strong and robust drivers (their success in rally races attest to it). One weak spot is the original SU fuel pumps; most have issues sooner or later, but a rare few claim they have never had one fail (routine maintenance is key, supposedly). Some have replaced the SUs with Facets or other brands, I carry at least one spare and can swap it out in 20 minutes; I like the sound of the SU pumps and I think it's part of the cache' of the cars (solid-state versions are available).
Most/all Brit cars are infamous for electrical issues, which is odd because the wiring seems, if anything, to be overbuilt, but I believe a lot of the issues are due to the ancient 'bullet-style' connectors (my dad's Model As used the same style!). Since you're doing a full resto I recommend a new wiring harness and connectors; and use dielectric grease in the connectors to occlude corrosion, which can be the source of many intermittent, frustrating 'gremlins.' Make sure to protect the wiring from any source of chafing or other damage.
Edit: Instead of dielectric grease, consider getting some of this, recommended to me by another owner. A bit expensive, but if you're rewiring the whole car it should be worth it:
The CP8-TB Kopr-Shield Joint Compound (8 oz) from Thomas and Betts is the copper colloidal surface treatment that protects, lubricates, and enhances conductivity between connections. It is an ideal solution for reliable connections in electrical and raceway installations. This unique blend of...
www.specialized.net
Some of us have fitted PCV valves; not a true PCV 'system,' but it helps reduce crankcase pressure and the notorious leak at the rear crankshaft 'seal' (the engine doesn't have a seal, but a scroll-type oil return on the crank, which relies on close clearances which, of course, wear over time). Some fit aftermarket seal kits, but a properly installed PCV valve can limit the puddle under the car. We fitted a seal kit to the BN2 it still leaks a bit (others have had more success).
The drivetrain is very strong; the possible 'weak link' is the overdrive, which is fun to switch on and off, and takes a bit of technique. Racers break them, but they're supposedly good for up to 200HP and in routine driving should go 100K miles or more without an issue (issues are often with the electrical system). Everyone seems to use a different technique to de/activate the OD; check back when you're ready to try it out.
I don't know if a '60 had a front disk brake option but, if not you might consider an upgrade. My BN2 has drums all around, but it's a lighter car and I don't drive it hard or for long distances, but the heavier BJ8 with front disks and a servo is the better stopper.
Sounds like you're a pretty good 'wrench,' so there won't be a lot of surprises. One possible one is the SU carburettors (Brit spelling), which are different in principle from American carbs; they're 'variable choke' as opposed to 'fixed choke,' but once you understand their theory you'll find they are simpler--with a single jet and no accelerator pump--and more elegant than American-style (with multiple jets, pumps, 'power pistons,' etc.), My dad wasn't a fan of 'foreign cars,' until he bought the BN2 which we restored together, but he rebuilt its carbs to perfection (he was a good wrench, too).
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