I went to Stonehenge without renting a car or going on a bus tour (ie, using only public transport). It's not that difficult, but like some have said, just a bit pointless esp since Stonehenge would only take half a day (at most).

If however, you do want to go to Stonehenge on your own from London, take the train towards Salisbury from London Waterloo Station. Once you have reached Salisbury, there is a bus outside the railway station that will take you directly to Stonehenge and the return ticket is GBP5, I think. While you are in Salisbury, consider taking a walk to the Salisbury Cathedral, which contains one of the four original Magna Carta. Salisbury itself is an old English town, so it's something you will not see in London, hence worth exploring as well.

If you want to go to Bath from Salisbury, there is a direct train from Salisbury station. However, I think Bath is such a beautiful town with so many things to do that it warrants a day tour on its own too.