To look for property to either buy or rent you can check on zoopla website. Nowhere in the area west of London / Heathrow is cheap although like most places there are good areas and not so good areas. The main things to consider would be noise from planes, commuting time and the possibility of flooding if you live near the river.
If you look at an aerial view of Heathrow you will see the runways run East / West. If you imagine a straight line from Heathrow towards Windsor and beyond they will be the areas most affected by plane noise. Obviously the further out you go the less you will notice them. Also if you go on the north or south side of Heathrow it is slightly quieter as the planes don't fly directly over these areas. Be warned that if Heathrow ever does get a third runway it will probably be on the north side, but this may never happen.
Your commute time will depend on where you end up living and what roads you use. The M25, M4, M40 & M3 in that area are always busy during mornings & evenings and even worse on Friday afternoons and if it is wet. This can cause you commute time to vary wildly even doubling your journey time on a bad day. I usually tried to stick to back roads at peak hours.
With regards to flooding, it happens rarely in this area but it does happen. Datchet and Wraysbury / Staines areas got hit quite badly during the last floods I remember about 3 years ago.
If you plan regular trips to London and will be taking the train (driving and parking is expensive and can be stressful in London) you should try to be near a train station on the main line to London (probably on the Reading - Waterloo line).
The nicer areas would be Windsor or Ascot but these are also expensive areas to live. If you prefer rural living more than town living you will be quite restricted but would suggest looking in the areas between the M4 and M3 motorways outside the M25. Train access will be more limited in some of these areas. If you go as far as Camberley / Farnborough or Bracknell you could be looking at a commute time of up to an hour with heavy traffic.
Are you looking to rent a house or a flat and how many bedrooms? I see you also have a dog which may rule out some rentals.
Have you booked any transport from the airport to your initial rental?
