Can I file a VFR flight plan for a training flight?
Absolutely. In fact it's not a bad idea to do so, if only to get some practice filing flight plans and using the system.

In the US VFR flight plans are a search-and-rescue tool: They're only used if your aircraft is overdue to give the rescue folks information to help plan the search.

In your particular case I would file a flight plan that lists your home airport as your departure and destination, with the airport you intend to use for your touch-and-go landings as an intermediate waypoint.  If you want to enter information on the nature of the flight (training, touch-and-gos at XXXX, etc.) in the remarks field you can do that too.

Activate the flight plan when you depart, and close it when you return to your home airport. You can optionally also call Flight Service over the radio when you get to the touch-and-go airport to give them a position report and let them know you've reached that waypoint (I would do this on your way out of the area to return home, along with an ETA update if appropriate).



As an alternative (or in addition to the VFR flight plan) you could get flight following from the ATC facility responsible for your airspace. Be aware that they may drop you and ask you to squawk VFR while you shoot your touch-and-gos since you'll probably be going off their radar, but you can always request service again when you start your return leg.