Speeds has already added most of the information.
You can easily find a good 3BHK flat with 1000-1200 sq.ft area in Vashi ,Nerul(Seawoods is a new node /part of Nerul ) for rent for 15-25k. (depending on the sector, proximity from the station,market, etc.)
Look for a place near Seawoods station ,if you plan to travel by train .It takes 23 mins to Chembur.
Seawoods is developing very fast and the upcoming station is a major project of L&T worth 1800 crores. one of the biggest in India.
Nerul has a lot of CBSE schools .Apeejay,DPS are one of the best CBSE schools in Nerul\Seawoods.
Vashi has more of State board and some new ICSE board schools.
If you want a peaceful and calm place to stay ..avoid Vashi .It has become very crowded and congested . but most of the malls and good recreational places are in Vashi.
Vashi and Nerul are connected by palm beach road and it takes hardly 10 mins by car.
Hope this helps .
Any other info you need, do PM me.
EDIT:
finding a place in Chembur to stay isn't easy ..and something like 1200 sq.ft under 25k ..very much impossible .

one of my friend who recently shifted to Vashi used to pay 20k for a 1 BHK flat in a NOT so prime area of Chembur.
EDIT 2 :

Firstly there is not much residential area near Vashi station Its all covered by malls,IT parks.
There are new completed projects on palm beach road (near vashi station) , most of them sea-facing towers and they all cost a bomb .That's like the most expensive area.
Depends on the locality (sector).
Sector 17 which is nearest to the station(5-7 mins walk) and one of the up market place in Vashi is quiet expensive too.
If you move a little away from the station , then comes Sector 14,9 etc. which is cheaper compared to Sec 17/palm beach.You might find a good flat in your budget there.
I would suggest you consult a few estate agents to find the best suitable flat for you.