Hello Orestia,
We moved to Kew two weeks ago, and we are really pleased with our decision. Like you, we also considered Richmond and Holland Park, as well as some other central London neighborhoods. With your budget, you can easily find a large house with a garden, close to the station, in fact there is one available on our street right now for 3000 and another for 2500. Kew village is mainly residential, which means that if you are commuting by Tube your walk to the station will be both short and pleasant. For us, this was a minor negative point about Richmond...Richmond is excellent for shopping, but generally speaking you have to cross or walk along noisy crowded roads to reach the residential streets from the station. But Richmond really is nice, too, especially if you can find something close to the Green or the Terrace Gardens. Something that I love about both Kew and Richmond is being next to the Thames, where we can enjoy walking along towpath. And you can buy a season pass for the Botanic Gardens, which is a good family activity.
Shopping in Kew is actually better than I expected, as it turns out. On your map you may see Kew Retail Park, between Mortlake Road and the river. The first time I went there was at noon on a weekday, and 90% of the other shoppers were women with small children. It was a bit overwhelming to see so many strollers, not something you will ever experience in central London I think. Evenings and weekends the clientele is more diverse, of course. Anyway, the retail park has Mothercare, Gap, Next (clothing), Boots (pharmacy) and Marks and Spencer (cheap department store and EXCELLENT supermarket). I was worried that without a car I would be doing most of my grocery shopping at the Tesco Express next to the station, so the Marks and Spencer was a pleasant surprise. Also, it is common for supermarkets to offer home delivery (for free, or for a small fee), so you certainly won't starve, no matter where you are. There is a huge Sainsbury's on Manor Road as you head down toward Richmond, and Richmond itself has Waitrose (my personal favorite). As for household things, I am a big fan of John Lewis department store which offers free delivery for any purchase over 30 pounds. (I need to figure out how to insert the pound sterling symbol, but I hope you understood that I meant cost, not weight.) So really, no problem not having a car! In any of these areas, Kew, Richmond or Holland Park, I am certain that you will be happy. If you have any questions about Kew in particular, I will be glad to try to help (bearing in mind that I am new here myself), so do feel free to ask. And four days with an agent will be more than enough...you will be spoilt for choice, and will probably find something perfect on the first day. Best wishes and have fun with your home-hunting!