Depending on the sign, you can use an industrial adhesive. We use Lords 406/19 but the distributor should point you in the right direction for your needs. However if you are looking for a very cheap fix, this isn't it. The last time I bought the applicator (a caulking gun won't work) it was ~$150 and the tubes are about $50 each plus mixing tubes. You WILL need extra mixing tubes because it hardens in the tube in a short time. I've known people to put together Signcomp electrical boxes with just this adhesive but I don't recommend it. Plus you need to prep the aluminum to get the mill coating off so the adhesive will bond properly.
A MIG welder is really great for aluminum. If you are just starting out, taking it to a welding shop might be best. A welding setup (with machine and spool gun) will probably start at $2500.00 - ignoring things from harbor freight which are IMHO junk. A good one is more like $4000+. IIRC I spent $3500 on my setup about 12-13 years ago.