I want to get from Ventimiglia to Antibes using public transport.
I know that the TER train costs about 10 Euros. Is there a cheaper way (a bus maybe?)
Public transit is often managed by local governments and therefore lines end at the border of their territory.
I suppose the train continues to Italy for international cooperation or historical reason (also simply because there is a need for public transit). But other transit systems don't go as far as Italy, buses are managed by the Departement des Alpes Maritimes, and therefore limited to France.
A ticket on this network costs 1 euro and not valid for transfer, so you'll have to buy 2.
The train from Ventimiglia to Menton costs 2,40 euros (full fare) so you can go down to a total of 4,40 euros.
Note: you will probably lose a minimum of half a day (it'll take 4h+) to save 5 euros. If I were you I'd take a look at discounts on trains. If you are under 26, you can have 25% off when travelling off-peak. If you plan to travel along the Mediterranean sea in France, you can take a look at the Carte Zou. It costs 15 or 30 euros depending on your age and you always have 50% off (so on a Ventimiglia-Antibes round-trip you already save 10 euros). It's not made for tourists but I guess you can use a fake address like your hotel's.
There is Bus 905 from Ventimiglia to Menton, but it works only during daytime
So the Itinerary becomes bus 905, 100 and then 200. The ticket cost is 1,5 euro each, so this brings the total price to 4,5 Euros