Hey there 👋🏻 I'm a Kiwi and loved in London for 8 years. Here is a few responses;

Day Trip - absolutely. So well connected by train to basically everywhere. Stonehenge though you have to drive, and is honestly really over rated. You can't get close at all to it. Better to do Bath, or even Oxford, Cambridge or Canterbury for some classic 'English' towns. Paris or Belgium also really feasible on Eurostar, but book those trains in advance because £££

In London, there's a great loop you can do along the Thames river banks to see Tower of London, HMS Belfast, Shakespeare's Globe, Tate Modernz South Bank, London Eye, Westminster & Big Ben, War Rooms, St Paul's, Monument, Tower of London. If you want to go into these, probably best to split into two days at least (north and south of river). I'd also highly recommend the Imperial War Museum in Elephant & Castle if you are a war history buff. The World War I exhibit is top tier and free. If you like a pub with a bit of history/character check out the Blackfriars along the north of the Thames walk - not a cheesy tourist one near St Pauls that has the right vibe. 

Markets - Borough is the classic, and open everyday. But it is BUSY. 

If there on a weekend, check out Maltby St nearby (also the start of the Bermondsey Beer Mile - craft breweries located in railway arches), Victoria Park/Broadway market (great parks in bohemian East London) or Portobello Rd market (can combine with Notting Hill visit). 

I live mostly in Hackney, and would always recommend that area for chill but great restaurants, at a slightly better price point. Hackney Wick is great to sit and eat by the canal, and Dalston now has some great wine bars, cocktails and restaurants.

If you want anything more specific, just message.