A tip I would suggest is to try and bring a few gallons of clean water from your old source with you. That way you have some water your fish are already used to to help acclimate them to their new water more easily. Old milk jugs (clean very well! Or buy a few water jugs at the store and just refill them with your tap water) are great for this if you don't already have something like a bucket with a lid or carboy that you can use.
I would suggest not feeding your fish the day before and while traveling, at least not the betta (I'm not sure of cory cat metabolism but most fish can go a few days without food no problem), to reduce the amount of waste they're producing while in their travel containers.
A tupperware container filled with tank water is a great way to keep your filter media wet during transport, you might end up having a mini-cycle because some of the bb have died off but it's better than starting from scratch imo.
To keep the sun off, make a "tent" with a blanket, towel, etc (light colored is better, or if you can swing it, one of those reflective car windshield covers). Don't lay it directly over whatever container you choose to transport your fish in but make sure there's air space underneath, and this should keep the worst of the heat off them. Of course, don't leave them in the car unattended for more than a few minutes, especially if you're parked in the sun- cars heat up fast!