Do I need to purchase/reserve seats on the TGV before arriving in France?
No, but you can save a lot of money if you do. Depending upon the departure time (peak or off-peak hours), the standard 2nd class fare ranges from 76.00 to 102.50 EUR.
When are you taking this trip? Booked well in advance (up to three months allowed) at www.tgv-europe.com, you can get a Prem's fare as low as 22.00 EUR. Prem's tickets are non-refundable and non-exchangeable so you have to be able to commit to a specific departure date and time.