You want to put the batteries in parallel, so you'll have double the capacity. Installing them in a series would increase voltage and most likely damage something (mixing 12V systems with 24V battery pack doesn't really work). Increasing capacity with another battery can cause strain on charger components, so make sure they're beefy enough or at least have proper protection against overcurrent.
People are correct that you should use the same capacity batteries, preferably the same make/model and age. Mixing batteries can cause problems where one battery drains faster and other(s) start to charge the lower level one so you'll have less useful amp-hours and that degrades batteries faster.
Switching batteries with a relay or a switch is possible, but you need quite big and good quality relays/contactors for that as current can be pretty high which can cause arcing and even weld contacts together eventually. With proper parts it's a safe way of doing it, but personally I'd just get two batteries in parallel since there's fewer components to malfunction and adding complexity with arduino+contactor doesn't save that much money, specially if you place any value for your time (which of course isn't necessary, tinkering itself is often worth the time spent).
Get a switching relay or one normally closed and another normally open one. That way there's no paraller connection at all. Connecting two batteries together with different voltage levels causes a huge current spike, think jump starting a car and how thick those cables are. Arcing will happen on the relay contacts no matter how you switch it if there's load connected.
And since you're not talking about a trailer, is one of the batteries for the car itself? Since if you're planning to use the secondary battery as a car battery too you need very heavy wiring to give starter enough amps to run plus running that over a relay is a whole another beast to manage since starter motor can pull hundreds of amps momentarily.