Welcome! I actually live close to Bromley - you'll find the commute ridiculously easy! An oyster card may be useful if you aren't getting too many buses or tubes, and there is a price cap when you reach the price of a daily travel card, but if you forget to swipe at a station you'll incur a charge so I'd personally just get an all zones travel card at the station. If you wait until 10am to go, then in will be £8.90 for the day, and this allows you to get as many trains, tubes and buses as you wish all day. Before that is classed as peak time so it'll be £13.something.
Firstly, for your day trip, how about Stratford-upon-Avon? Shakespeare's birthplace! It's a bit far away but it's really interesting!
If you're in to shopping, try Bluewater - it's actually in the opposite direction to London, roughly 30 mins from Bromley? But it's the largest shopping centre in Europe (or at least it was when it was built lol!) It's built in an old chalk quarry which is pretty cool
I'd check out Camden - quirky stalls, street art and a 10 minute walk from London Zoo
You could even just walk the streets and people watch - you won't find a wider variety of people anywhere! If you do decide to venture out at night, Koko usually has some good live music on and Purple Turtle isn't bad.
China town is always good for some chinese food and some odd sights, such as whole octopuses hanging in windows.
You could also have a wander around Borough Market - it's all food, but you can get some really interesting stuff there. There's a stall where you can buy odd things like kangaroo and crocodile to try! And of course, you can see the place they used as the front door for Bridget Jones' house
Tate Modern's always worth a look, as is the Tate Britain and National Portrait Gallery. The Science Museum is pretty cool too.
Covent Garden's a nice place to walk around, do a little shopping and there's always some sort of street performance going on.
You could see the Cutty Sark.
Greenwich Park is beautiful and will be easy for you to get to - they have deer! (This is somewhat fascinating to me, as I had never seen a live deer until I went there lol) You also get a really beautiful view of the city from there.
If you're interested in theatre, The National Theatre usually has some reasonably priced shows on, or if you're willing to spend a little more go West End! We Will Rock You and Wicked are my personal recommendations
You could go to the viewing platform at the Shard.
You could take a ride on the cable cars that go to the O2 in Greenwich (once the Millennium Dome) to get a great view of London without having to spend a whole hour on the London Eye.
Madame Tussauds or the London Dungeons are fun if you're in to that sort of thing.
Basically, there is no shortage of things for you to do and see here! I would pack a mixture of clothing. As it's July, you'd expect the weather to be pretty good, but rain is always around the corner lol. It shouldn't be too cold though, so just make sure you have a couple of jumpers, jeans and a lightweight jacket. And something summery in case you luck out with the weather
Hope you enjoy your trip!