I can just describe what I used for those tasks.
curl - if you are lazy, you can just download in built binary and have nothing to do more, it has very simple headers and a lot of examples. You will need 3-4 already built libraries and header. With no external dependencies. So, I would count it as too low-level, but not heavyweight at all.
boost.asio - very interesting paradigm of realization, quite easy and clean. But I would say it is low-level too. Harder then curl. And needs to use boost, that is external dependency you want to avoid, I guess.
poco - best solution for http server. It's high level, after you joined it to your application, you just need to implement few virtual functions, having all other work (and thread management) done. Poco gives a lot of stuff for application management, thread/process management and it is very simple and easy, I would even say it has java-like interface. Yes, there is external dependency to poco, but I would look at this as a chance to learn perfect library. Still, for http clients it is really heavy.
That is all my C++ experience with HTTP. Counting you need just client and don't need dependencies, I would offer you to look at libcurl library. It's cross-platform, easy, no dependencies and low-level enough to get all you need with network. And if you will have time - look at Poco, really, I believe you will fell in love with this library as I did.
Hope that will be helpful.
