Kew Gardens

Take the Piccadilly Line from Russell Square to Hammersmith, then walk across the platform and take a District Line train in the same direction. Make sure you take a train with the destination "Richmond", no others go to Kew. Get off at Kew Gardens station, the Victoria Gate of the Gardens is within 5-10 minutes' walk and well-signposted. Kew Gardens is in Zone 3 travelling from central London, make sure you have enough Oyster PAYG money or a Travelcard that goes out that far. If you go back into town the same way, make sure you get an Underground train and not the Overground that goes in a loop round the suburbs.

Emma, sorry to correct a fellow Londoner but that's wrong. The train from Waterloo serves Kew Bridge station which is nearer to the old Kew village centre and a significant distance from Kew Gardens (maybe you mistyped). Also, I think that for the OP to get from Bloomsbury to Waterloo will be more of a hassle than getting the tube all the way.

While we're on the subject, any thoughts on taking the Thames River Boat back from Kew to London?
http://www.thamesriverboats.co.uk/kew.html
We're going in late May and (weather depending) we were thinking of taking this on the way back.

Thanks to all for your comments. I am familiar with the underground and have an Oyster card from past visits. I like the Picadilly route from Russell Square as my hotel is just down the street from RS. Now how about some nice early June weather?

My husband and I have taken the Thames River cruise twice. We go out on the long boat ride and take the train back into London. I believe there is just one boat company which does the long trip out to Hampton Court. We had to take the earliest Thames River boat available, in order to have time to come back on the train. We totally enjoyed that particular Thames route. We have also gotten off at Kew Gardens from that tour company.