It took me some years of travel to London before I arrived at Greenwich. Before that I stayed in the city or in the north of London to be closer to the airport. However, staying in Greenwich you are still close to the city's hot spots but could experience the more village type atmosphere along with the real flair of Great Britain. Things the traveller normally likes to feel of a country.
Besides it's easy to walk around, not that crowded and hectic like in the city's centre. Still giving room to relax on your trip.
If you fancy public transport and you should in London get the "Oyster Card" The London Oyster Card is a rechargeable travelcard that can be used on transport for London. If you buy a London Pass + Travel package, this will include a sightseer credits package (London Pass) and a pre-paid Oyster Card (travelcard). The card might be also bought online or even upon arrival at the airport.
To reach Greenwich from Heathrow airport you take the Picadilly Line to Green Park, change there to Jubilee Line until Canary Wharf and then take the "DLR" train to Lewisham.
All within walking distance:
A variety of smaller shops are in the (city) centre with typical British goods. There is also a flee market with food stalls on weekends (come Saturday). For the modern lifestyle passionates : Canary Wharf Shopping Complex is waiting for you. You passed it already on your way from the airport.
Canary Wharf is a modern business complex. The attached shopping center is really huge and offers the classical brands as well as smaller shops, food, restaurants, super markets. I found it a great place and very easy to get around within the building. You get good bargains as well as up market clothes.
Last time I stayed in the Doubletree Hotel by Hilton which might sound more expensive as it is. You must not necessarily go in the summer holiday season ;-)
The Hotel is near to the train station (5 min. walk) and the City Centre is 10 min. walk away. Service and rooms are good as well as the breakfast which got ***** rated by me! And I am not getting paid for these comments :(
In case you are staying 3 nights you should have enough time to take at least one trip on the river Thames from Greenwich to London which is a nice experience and provides good photo opportunities. Tickets could be bought at the pier.
Along the Main Street towards the river you shall find a variety of Pubs and Restaurants. However, you might also have a look in the 2nd line streets - there prices are a little bit cheaper - like always. I have tried 2 and apart from drinks and beer they are mostly serving dinner a la carte. In any case much cheaper than in the city! The Mitre Hotel & Restaurant could also be recommended here. It is situated at the Main Street and can't be missed.
More info? Just drop a note!