The Tube can be a bit overwhelming for people that have never used a system like this but once you ride it a few times, you will get the hang of it (my parents have!). First, you need to know where you are going in order to use it. I have a mini map on my phone and in my diary(planner) which I use to get around.
Remember to stand on the right and walk up the left of the escalator to avoid frustrating other passengers.
The best way to save money using the tube is to get the Oyster card. You have to pay a £5 deposit to get the card and you can put some money for your travel on it like a prepaid card. The reason why having the Oyster card saves you money is because it only charges you for the trips that you take and no more than than a travelcard.
For example Jane and I took the tube from Kings Cross to Westminster for £2.00 and Green Park to Kings Cross £2.00. So we saved £3.00 on that day. The good thing is, it will never cost you more than a travelcard but can cost less!
Some people buy a travelcard but hardly use the tube as it can be easier to walk and you miss the sights whilst underground. If the Oyster card wasn't the best way to travel by tube, you would not see all the locals using it. You can also set it up to Auto-top up if you go down to London frequently so you don't need to top up when your card is low.
In January 2015, they changed the ways that the fees for Travelcards work. They got rid of the Travelcard for Zones 1-2. If you travel in by train into Kings Cross or Liverpool St, you generally stay in Zone 1 sightseeing. If you get a Travelcard for Zones 1-6 you pay £17.00 (£12.00 off peak) but the Oystercard you will never pay more than £11.70 and if you only stay in Zone 1, you only pay £6.40. That is almost half the price of the Travelcard. If you drive to Epping, you will need a Travelcard for Zones 1-6) or Stratford (Zone 3), you will need one for Zones 1-4 (£12.00).
There is no way that you can convince me that the Oyster card is not the way to go. When you leave, cash in your Oyster card and get the deposit and balance back. Read more about it here.