Family of 4, what would be the most cost effective way to get from LHR to either of the station
The Piccadilly Line goes directly from Heathrow to South Kensington. If you plan to go to Victoria, then make your connection to the District Line at Hammersmith.
Use the tube. Get the Piccadilly Line from Heathrow to Hammersmith. Cross the platform to take a District Line train in the same direction to either Victoria or South Kensington. (They aren't that close together - there's also Sloane Square station in between.)
With the Oyster card it should cost me 2.20gbp or would it chare again for changing to district line
Heathrow is in zone 6 and both South Kensington and Victoria are in zone 1. The Oyster card single fare from anywhere in zone 6 to anywhere in zone 1 is £5.00 before 09:30 on weekdays and £3.00 after 09:30 on weekdays or anytime on weekends. You can make unlimited transfers until you exit the system at no additional charge.
We took the tube from LHR to S. Kensington station without any changing and the cost was low-much less than any Connect train. After we got to SK station we pulled our luggage around the corner to our hotel, The Rembrandt. Very easy. We had planned to take a taxi but it was not necessary though it may be needed for your group depending on your hotel location.
In your other post you said you had found an hotel near Pimlico tube station. If that is still correct, then take the tube from Heathrow (Piccadilly line) to Green Park and then change to the southbound Victoria line. You really won't gain anything by using Heathrow Express/Connect to Paddington (you will waste some money though).
That is correct, I did find hotel near Pimlico tube station
You can go on-line and hire a pre-arranged, private, shuttle service. That is what my husband and I do from Heathrow. We have also used the service to get from our hotel in South Kensington or Westminister over to San Pancreas or back to Heathrow. The driver meets you right when you pick up your luggage, helps with the luggage and walks you to his car. You would need to tell them you are a party of 4, so you would have enough room for four people, plus all your luggage.