Right in that area there isn’t much, closest would be:
Hunter Farms in Princeton - it is a nice facility and I believe offers lesson on school horses. However, I do not know how much turnout they have, if any.
A fairly short drive from Princeton is Duncraven (although by short, its probably 30 minutes). Duncraven is a really nice facility run by great people. I do not know if they have a lesson program, but definitely worth checking out.
I believe that there is another facility close to Duncraven called Quiet Victory Farm. I don’t know anything about it, but it looks nice on the website.
Hidden River Farm is in Neshanic Station - good reputation and nice facility. I do not think that they have a lesson program though.
There is also Hay Fever Farm in Robbinsville which is owned by the Shapiros. Looks really nice and other posters have commented positively on it.
There are also a lot of options in Hunterdon County and Somerset County - but you are probably looking at more of a 45 minute to hour commute for these places.