Hey
I just got back to London. It's cold and cloudy and sometimes wet here at the moment. This is typical because this weekend is a Bank Holiday weekend.
That is possibly good news for you if you want to do carboot sales and such, because there are some that only run on bank holidays. I know an amazing bank holiday carboot that is probably running this weekend/monday, weather permitting, but getting to it requires getting up very early, getting a train and then walking for 15 minutes.
It depends where in the city you are staying, though, as to how easily you can get to a carboot sale. There are some in the city on a Sunday which run every week regardless of the weather. I have two that I go to on and off, both of which can be reached by bus or possibly tube but it depends where you are coming from. If you want to PM me I can try and give you more details about those. You don't need a car to get to them. In fact, in central London, having a car is a hindrance, not a help.
London transport tips:
There are two types of Oyster Card (travel card) you can get here. One is unlimited travel over a short period, the other is pay as you go. You can probably get this at the airport, but I am not 100%. Big rail stations also carry them. I don't know if you are coming into Heathrow or Gatwick (I'm assuming one of those).
Heathrow to central London - there's the Heathrow express (Most expensive option, also quickest, 15 mins) which is a direct train into Paddington Station in the (sort of) centre of London.
THere are also local trains that run the same route with a few more stops so takes a bit longer but costs a bit less. You can get tickets for both of the above at the airport.
Thirdly there is the tube (the underground/subway). I did this yesterday from Heathrow home and the train was packed, because they had an issue, but most of the time it works. The only thing you have to look out for with the tube is that on weekends sometimes they suspend part of the service for maintenance. The line that runs from Heathrow (all terminals) is the Piccadilly Line (dark blue) and it goes right through the middle of London - Leicester Square, Piccadilly, Covent Garden, etc, to King's Cross/St Pancras and then beyond. You can change lines at some stops along the way. That is probably the cheapest route but it can take 45mins to an hour depending on your destination.
Gatwick is further out - there are trains (Gatwick Express is the quickest and most expensive, about 40 minutes though?) and Southern Rail run local services that stop more and so take longer but are cheaper. There are no tubes from Gatwick.
There may also be buses from the airport.I don't know about these.
Is the tube safe? As safe as any transport anywhere in Europe can claim to be at the current time. If you are travelling at rush hour, it's not the friendliest of services. But the tubes do operate a good part of London, especially in Zone 1. If you are using the tube, try and get in the front or the last carriage. Avoid the middle carriages, at busy times you will get squashed.
I personally prefer the buses, though, because they are cheaper, and they go to more places. They do take a little longer in some cases, but you get to see the city as well, which you don't get with the tube. There are a lot of London buses, some run up to every 3 minutes, and most are doubledecker buses. Again, which buses are most useful to you will depend on where you are staying.
Touristy sites - depends somewhat on the weather, for example the parks are pretty but if it rains, they're not so pleasant to visit! I haven't been a tourist in London for a long time, but some options include:
Tower of London (possible in rain or shine) - tube to Tower Hill, or bus.
Westminster (very busy at this time of year. This is the area where there was the recent attack, but it is a very busy place for tourists and (touch wood) that is the first time something like that has happened there. Tube to Westminster or bus...
Museums - British Museum (Tube: Russell Square, Bus: British Museum, route 10). Natural History Museum and V&A Museums - (Tube - South Kensington. Can't remember which bus route off the top of my head)
Trafalgar Square (tube to Charing Cross, or bus - I know routes 91 and 29 go there, there's another one that also goes through Westminster but I can't remember it right now)
There's also Harrods (Tube to Knightsbridge) or Hamleys (Tube to Piccadilly I guess, buses stop right outside it though). Hamleys is a toy store worth visiting, though very expensive to buy!
I am sure I have forgotten lots of stuff, I've been in Japan for three months and like I said, I haven't been a tourist here for ages xD.
Pony shopping - if you want newer pony stuff, I would advise you to avoid any shops in central London whatsoever. It does depend on your location, but stores like Smyths and the Entertainer are not right in the heart of the tourist area. There is a very odd TRU at Queensway, but it is odd, so. Entertainer has a branch in Shepherd's Bush (tube/bus will get you there, bus will take a long time though) and one in Stratford (both in Westfields shopping centres I think). In terms of Smyths, the other best place in my experience - those are out of the city and you'd need to get a bus to the wider zones to reach one. I only know of the one near me, and I live in zone 2, and it's a 45 minute trip for me...so keep that in mind.
Second hand pony stuff - aside the carboot sales, very rarely, I've never seen any G1 stuff here.
Food...I don't do restaurants, so can't help you there.