Tube is cheapest and overall likely to be fastest though if you're arriving in the morning that's fairly academic. Piccadilly line to Hammersmith, change to District line for Westminster. If you're staying at one of the County Hall hotels, take exit 4 Bridge St, turn left and walk over the Bridge.
You can get Oysters at LHR, load with travelcards, PAYG or both. Which is best will depend on how long you are here and whether you want to use the 2for1's or not. You can also use contactless cards or Apple/Android Pay for travel - works the same way as Oyster, if you have these.
If you tell us exactly where you want to go, please, you will get the best directions. The advice above is good.
The Tube is usually the fastest way to get from Heathrow to wherever you're going in London. If there is only one of you, or two of you, then it's definitely the least expensive method of travel. What is best is subjective and depends on your date of arrival, time of day, number of people, amount of luggage, any mobility or other physical issues, et cetera. Other options include a pre-booked car service (possibly good value depending on the number of people and where you're going) or a taxi off the rank at the airport (most expensive).
If you want the 2-4-1 discounts, you can't use an Oyster card for those, so that may direct your decision. This also depends on how many are in your party and so forth.
Again, best depends on the details, especially length of stay, number of people, and if you want the 2-4-1 deals, so please give us more information so that you will get better advice.
My husband, son and I are coming in from Edinburgh midday on a Friday into LHR. We need to get to our AirBnB apartment, located on Whitehall Ct. We will have luggage with us. Is it easy to travel on the tube with luggage? What about the Heathrow Express? I know it's a busy time of year. Would private car be better?
I don't know anything about 2-4-1. What is that and how does it work?
The tube won't be as busy as earlier in the day at peak times. You shouldn't have problems getting a seat, they run every 10 minutes or less so if too busy then wait for the next one.
A tip for Hammersmith, you will exit on the left side of the train looking to the front so try to store your luggage on that side by the central doors.Edited: 4:02 pm, April 04, 2017
You will be closer to Embankment than Westminster tube station for day to day travel, although there are a few steps so Westminster might be the better station to arrive/depart with heavy luggage as it has a lift.
Can someone please explain the 2-4-1?
"'m still trying to figure out whether to get the Oyster Card or the Travelcard"
They aren't mutually exclusive (a 7 day travelacrd from a TfL vendor will come loaded on an oystercard). The only reason to get a paper one from a national rail outlet is if you need it for 2-4-1 entrance deals.
A 7 days travelcard is likely to save you money and be more convenient than other options.
Although I'd question whether you really will be travelling 'all zones at all times of the day and night' - you'll either be doing a fairly non-standard trip or you have lots of plans with friends that live in the suburbs. Most tourists would only need a zone 1 and 2 travelcard with some extra money added on an oystercard for excursions outside those zones, but you would need to do the maths for your situation.