Of you prefer an even more entertaining approach and a very gentle introduction into the topic, I recommend the comic "Logicomix" which tells Russel's journey (though not historically correct all the time for story telling reasons).
Logicomix is novel, and done well, but flawed … it's deficiencies lie in what it leaves out which may come across as an unfair charge but in this case the charge is warranted. There is a more historically correct and less orthodox work waiting in the wings for whosoever should attempt it.
It's also a bit weird how it makes ahistorical meetings for "drama" -- e.g. Russell and Frege never met in real life -- why show that they did (they do mention this never happened in their notes, but why write such a scene at all?)