All the big three Canon, Panasonic and Sony will have cameras in the $300 budget.
My choice would be the Panasonic HC-V210.

Remember you will also need a SD card, various cabling to connect to your TV. Plus nice to have, extra battery, an editing suite for your up to date PC and blank DVDs or other storage media.

Recently I amalgamated all my video of my Granddaughter and made a 0 years to 2 years compilation.