I would say ask your doctor first,
But I've had problems with my shoulder for years ever since an injury, and my recent dedication to weight training has made my shoulder problems disappear.
I'd ask your surgeon and physiotherapist what is best for you and stick with the shoulder/arm exercises they gave you. Walking, running, exercise biking, cardio, etc. should be OK but I'd watch any weights or push-ups on that joint!!! - Especially any overhead workouts.
I had my 2nd reconstruction a couple of years ago and I'm still very limited with what I can do. My surgeon told me post-op that I am never allowed to put my arm above my head again (yeah, umm, OK, but I do when I need to!).
I have very 'bendy' joints and hyperflexion (I can bend most of my joints past the 'normal' limits...yep, right into semi-freak zone!) My first surgery was fine and the joint lasted about 10 years until I fell down one night and a friend grabbed my arm and pulled me up....errrr, POP!
The shoulder was good for another couple of years until I started gentle swimming to get fit. I did 3 laps of freestyle in our backyard pool and the arm literally just fell out with no pain and no warning! - That one turned into a disaster as the Emergency Dept thought it was a great idea to let a trainee doctor put it back....if I wasn't so doped out on pain meds I would have said no, but I didn't! Without scaring everyone else here or going into detail, I literally felt things ripping. - Two weeks later, I was in for the 2nd op. I had so much damage from numerous dislocations (from the decade before) and from the young ER doctor wrenching on my arm the wrong way that the ligaments and tendons were rather ripped up. :(
My surgeon showed me photos he took during the op and he did some fancy stitching to reattach it all. - Anyway, so now I'm limited to wall push-ups instead of floor ones and even sit-ups are a no-go for me.....but I do very gently do some! My surgeon told me that if I dislocate the shoulder again, that I'll be back in for a bone graft (err, no thanks).
So my surgeon gave me a list of various exercises that are designed to strengthen the joint without causing pain (or damage) and my physio taught me how to do these correctly - that is the key. Honestly, I am still building up strength in my shoulder (the other one isn't great either, due to a benign tumour, but that's another freaky story!!!) and I put on a heap of weight after the last op. I started with very gentle elastic band exercises for the shoulder and have worked up slowly. I am currently renovating (ripping down plaster, hammering, drilling, smashing tiles, plastering, sanding, painting....everything my shoulder hates!) - but I still need to be mindful and careful with my shoulder.
Cold packs are your best friends post-workout! If the joint is feeling hot or a bit 'grindy' - rest it and pop a cold pack on it for 15 minutes or so.
Hope this helps! Just be sure to give your body time to heal - don't be too quick to jump in and make your recovery longer! :)
I had a similar surgery 3 years ago. It wasn't until year 2 I really used proper weights. My shoulder now is mostly artificial now, so I can't feel like in my other shoulder.
Back to your question... start off with hand weights. I went 5lbs to 8 to 10 and increased over months. MONTHS. Don't rush back into it. Be patient. Or you'll end up back to square one.