London Pleasure Gardens has programmed events for the entire summer, many of them with free entry. Check the website for more info.
Note that although the festival is free, you can't take in your own food and drink (so not so 'free' then!). Also, check tfl.gov.uk for train times since it appears that every alternate train is re-routed due to engineering works this weekend.
A visual of the London Pleasure Gardens site taken from their website.
Another way to get there would be via cable car from North Greenwich with a 15 minute walk at the other end – as Deptford Misc points out, the cable car stations are not very conveniently situated – but it could make for an interesting day out, especially if you cycle.
If you're looking for excitement, it sounds as if the cable car trip has the potential to rival any ride at Thorpe Park, since if it's windy, the cars stop and you may find yourself suspended for some time over the Thames – see 853blog's latest post.