If you are looking to really clean the undercarriage, then a pressure washer (at home or in a wash bay) is a good option. I would do this every 2-3 weeks in the winter. Road salt is pretty harsh but the chill of winter generally keeps the chemical reaction slow. Water accelerates things, as mentioned already.
If you are looking to keep the body of the car looking good, you should really do some research on Optimum No Rinse. With 1-2 gallons of hot water in a bucket, you can wash your car in <15 minutes in your garage. Doesn’t get easier than that and the results are great, especially if you use good wash technique. Optimum now has an ONR variant that has their car wax in it (which is awesome, BTW). Definitely try a rinseless wash out. This is modern technology at its finest, and may take some getting used to.
I would try to avoid automated washes if possible. The ones that use mechanical agitation WILL DEFINITELY swirl your paint. No doubt about it. The touchless washes use very harsh chemicals and will dull exterior rubber/plastic pretty easily. If you have a permanent sealant like Opti-Coat on the car, touchless washes are a good option.
Summary – if it were me, get a pressure washer and hose the nooks/crannies/undercarriage of the car off every 2-3 weeks. Do an ONR wash on a weekly to every 2 week cycle. Make sure you have a good sealant or wax on your paint. If you want something easy, use ONR with wax or buy some Optimum spray wax (it is great stuff) and spray it on monthly as your are “drying” the car after an ONR wash.