Just so you know, the streets people have recommended you visit aren't in Dulwich Village, they are in East Dulwich. If you want to check out both areas, I'd suggest the following walking route.
I really feel for you as it can be hard to check out an area you don't know without any guidance!
If you are heading back to the area for another walk around, I'd suggest you check out East Dulwich and Dulwich Village. Start at Goose Green and walk up Lordship Lane. Turn onto North Cross Road (which has a nice Saturday market) and then go back onto Lordship Lane and walk up until you get to Townley Road-after Townley Lordship Lane is largely residential with a few exceptions so you can skip it. Take Townley to Dovercourt Road and then turn on Court Lane and head into Dulwich Park which is lovely.
Leave the park via the entrance by Dulwich Picture Gallery (ask if you are unsure as everyone will know). The Picture Gallery is nice if you want to visit it but if not, just head north down Dulwich Village road (this is the Village High Street) toward North Dulwich station.
Just as a heads up, on Lordship Lane there are a few restaurants that are changing hands at the moment so even though they may look closed, they are actually being refurbished and taken over. Franco Manca is opening near the Bishop and due to open in December, the Adventure Bar is being taken over by Tsunami (the Japanese place on Charlotte street and are due to open later in October), the old Church Hall is being taken over by the Picturehouse indie cinema chain and is scheduled to open in December and the former Speedo Pizza site near Boulangerie Jade is being taken over Burro e Salvia (from Shoreditch). As someone else previously mentioned, M&S just had their planning application approved to take over the Iceland on Lordship Lane so there are a lot of changes in the works. Also the Greyhound Pub in Dulwich Village is being turned into a boutique hotel so that is currently shut for works. When I actually list it all out, it is kind of insane how many places are changing at the moment!
If you have time for a detour, I'd recommend you also do the short walk between North Cross Road and Whately Road on Crystal Palace Road while you visit the market as there are two popular pubs (the Actress and the Great Exhibition on that stretch).
The fitness places can be hard to spot but there is a large Fusion on the bottom of Crystal Palace Road and East Dulwich Road in addition to ESPH gym and Push Studios which both offer yoga, Pilates, Zumba etc. Those studios are on Lordship Lane near the gelato place Oddono's. If you aren't too tired by all of that, I would jump on a bus down Lordship Lane towards Forest Hill and visit the Horniman Museum and Gardens if it is a nice day.
Due to the Saturday market on North Cross Road, I'd definitely recommend that as the best day to check out the area.
I hope you have better luck with your second visit and your general search. Even if you live in Camberwell, you can walk to most of the amenities in Dulwich listed above in less than 10 minutes so that might be the right compromise in the end.