Healey Friends:
We travel quite a bit with old cars and plan to take our Healey from NM to Vermont this summer. We don't carry many spares or tools, we just don't have all that much room. We do take little items that can be stuffed into corners, like ignition parts, bailing wire, hose clamps, or an extra fuel pump. Our 100 is miserable in rain, so we carry rain suits, RainX, chamois for cleaning the inside of the windshield and cups for bailing water out of the car.
We don't go anywhere without our shop manual, a recent copy of Hemmings, and rosters from all the car clubs we are in: especially the AHCA and AHCUSA. Finding nearby old car nuts and phone numbers for vendors can be very helpful if you have problems. We have a current AAA premium towing contract (includes 100 miles of towing and an 800 number to use it), though I have only had to use it once.
I find traveling with an old car is about the most fun you can have with them. You will meet new friends, have adventures, and learn a lot about your car. And your Healey will be a better car during and after the trip when you fix all the big and small problems you discover.