I have bought a fridge and washing machine in the last 2-3 years.
First thing I'd like to tell you, a lot depends on luck. Different people will have good and bad experience with all brands. So don't expect there to be one single best brand which never has any issues.
Washing machine, I'd recommend getting a front loading machine. Reason being that they use less water and clean your clothes better. They don't damage clothes like top loading machines. However, this might not fit in your budget. You can check bosch and IFB's 6 KG machines. This will work for 1-2 people. Wait for sale and avail card discount, if possible. Might bring the price within your budget. I have IFB, running well so far, after 29 months.
For fridge, I'd recommend LG or Godrej. LG has good after sale service. At your budget, you'll only get single door ~190L, which should be fine for single person. However, I'd also recommend looking at Rentomojo. I used their single door fridge for one and half years with no issues and bought my own before I got married. Refrigerator rents are pretty low, that's why I recommend. Washing machine rentals are high and not really worth it unless you plan to move very soon.
Lastly, for AC, I'd recommend you wait and see if you even need an AC. I have relatives living in Bangalore for 10 years and they haven't felt the need to get an AC in their house so far. The weather is fairly pleasant for 9 months a year. If you do feel the need for an AC, you can once again, look at Rentomojo first before deciding to buy. More than the cost of the AC itself, the hassle of getting it fitted and removed is something you will have to worry about, if you're going to live on rent. Moving an AC can be costly affair, can cost in the range of 6-7k. Rentomojo can save you lot of money here.