Newbie Non PAL rifle

I am looking at getting myself and my boys (12 and 13) into this hobby.
The rifle would be used for mostly plinking/target shooting but I would like something to control pests in the garden, rabbits in particular, once we can shoot accurately.
From my browsing of forums this hobby can get addictive and expensive.
To start out I have a budget of $300.00 or less.
Any advice on rifles, care of, techniques, from the community would be much appreciated!

Are you leaning towards spring driven rifles or CO2/AIR?

Leaning towards "spring"