If you have purchased a London Pass with an Oyster Card, I see they have recently changed what you get with this, so it is now a better deal.
With a 3 day London Pass, you get an Oyster Card pre-loaded with £25 credit. This is a Pay As You Go card, abbreviated PAYG.
If you want to take the tube in from the airport, you just follow signs at Heathrow for the Underground. You tap your Oyster card to the sensor to open the gate ("tap in") and ride to your destination, making any needed line changes along the way. When you get to your final station, you tap the sensor to open the gate ("tap out"). The correct fare will be deducted. Since Heathrow is in zone 6, and your hotel will probably be in zone 1 or 2, this fare will be higher than a fare within the central London zones 1 and 2. Furthermore, fares vary not only by zone, but also by time of day (peak costs more than off-peak). But don't worry about it - the card will automatically calculate the fares, and will apply "capping," meaning once your reach a certain maximum fare for the day, it won't charge you any more.
If the £25 credit isn't sufficient for your trips, you can "top up" the card (add value) with cash or credit card at a machine in the station. Almost all stations have uniformed helpers who can assist with this.
There are a gazillion hotels near various central London tube stations (yes, that is the precise number). As said above, do not confuse a tube station with a train station. If you were going outside of London, you would need a train station, but be careful - London has many of them, each serving different destinations. Some destinations are served by more than one station. So even if you are taking a train, you need to know which one if you want to be near "the station."
Depending on where your hotel is, the tube may or may not be the best way in. Or, you may simply prefer another method. For instance, if there are several of you and you'll be tired and/or you have a lot of luggage, you may prefer a minicab. That's Brit-speak for a prebooked car, which is much cheaper than getting in to a black cab at the airport. If you want to take the Heathrow Express, note that the tickets are quite cheap if booked 90 days in advance, and extremely expensive if bought on the train, and other scenarios are in between. Furthermore, while it is only 15 minutes, it only goes from Heathrow to Paddington Station. Again, if your hotel is not near Paddington, this may not be a good method - at any price.
If you want more specific hotel advice, you have to give us details. How many people, of what ages, do you want per room, do you have any specific needs/wants (elevator? air conditioning? view?) and what do you consider "medium priced." In the meantime, I recommend looking at the Premier Inn chain, which has lots of hotels all over London, and is a good value. It's where lots of British people stay; if your goal is to "meet locals," look no further than a Premier Inn breakfast room!
For itinerary ideas, start with Rick's London guidebook. Also look at the London Pass's complete list of covered attractions; you may spot something you haven't heard of that interests you (this is how I learned about the London Jewish Museum and the Museum of Brands, Packaging and Advertising). If you want more specific advice, you will again have to give us details. "What should I see with three days in London" is too general (that's what guidebooks are for). But "I want to learn more about WWII" or "I'm interested in buying used CD's" or "Which stores in London would have scrapbooking supplies?" are things that can be hard to get from a guidebook, and where a forum like this is useful.
