You need to test both the battery and the alternator. Does the battery have caps so a hydrometer can be used to test specific gravity of the acid? That is the only true test and it is dangerous because of the battery acid- it will burn and blind you- use eye protection- no joke. That is the usual battery test but it assumes you have a fully charged battery to begin with. That is why it is a good idea to have a charger, just running a car is not designed to charge a weak battery, only to replace the charge that was used to start the car. Of course lots of running is better than a short hop, but an overnight charge is the best thing you can do. Then, once the battery is fully charged, and tests OK, you can test the alternator. The carbon pile tester is the one to use, it puts a load on the alternator and forces it to really put out. A good one is not cheap but Harbor Freight sells a cheap version that is OK for casual use and most car repair shops use one. Maybe that car club has one. The battery acid tester is not expensive, but if your battery does not have caps, and is sealed, you are out of luck. After all that, then if you have starting issues, time to look at the starter. Aren't old cars lots and lots of fun?