do not pay for tourist buses. buy a cheap guide to london and read up before you come. remember that an oyster card/capital card is the same as a hop on hop off bus. you can travel on anything for the day.
charing cross is a good place to start if you are going to Greenwich. you can take trains out to London Bridge and then Greenwich (you may have to change but it is straight forward. from the station you can get to the park in 5 minutes. lots of signs. getting back into town you can take a boat or the docklands light railway to get a good view of the docklands area and the new financial centre. the museum of london is at westferry dlr station (i think).
from charing cross you can walk to trafalgar square (one minute) then admiralty arch, the mall, buckingham palace and st james park (5-10 minutes.
from charing cross you can walk to st martin in the fields, the national gallery on trafalgar square, the portrait gallery (both free for some parts).
the theatres are here, lion king is showing at the bottom of the strand, turn right out of charing cross, but the tickets are very expensive.
if you turn right out of charing cross station and walk for about 5 minutes, then cross the road and turn left, you will get to covent garden. there is the london transport museum and other stuff.
you can walk from charing cross down whitehall to horse guards parade (you might see soldiers on horses, or on foot standing guard) downing street, big ben, the houses of parliament, westminster abbey the wheel thing on the river (if you turn left at the bottom of the road). if you turn right, past parliament square you can walk to westminster cathedral (the catholic cathedral) in about 30 minutes.
by bus from charing cross, take the 11 it goes past everything above, or the 24 just in front of south africa house on trafalgar square. both buses go to victoria station.
charing cross, buckingham palace, houses of parliament can all be done in one morning.
from charing cross, you can walk over the river to the south bank and down to london bridge if you have the energy, passing the national theatre, tate modern, down towards tower bridge and city hall.
do not pay for a tourist bus. your oyster card/capital card will cover everything for a fraction of the price.
food, print off vouchers for chain restaurants like, pizza express etc but you can't use vouchers on the south bank.
there are lots of boots, m+s, tescos, sainsbury's coops to pick up cheap lunches.