I just finished writing a DLL that wraps latest libcurl so I can use HTTPS with Delphi, Visual C++, etc. without bundling 3rd-party DLLs.
Unfortunately, I needed to add client-side certificates so both the server and client can verify each other. And I needed the certificates to be read from RAM. I was under a lot of time pressure, so I used Poco instead of modifying my libcurl wrapper.
libcurl's easy API is indeed easy, but it does things in C (no classes or streams etc.):http://curl.haxx.se/libcurl/c/libcurl-easy.html
Poco uses C++ with classes, iostreams, etc. and I prefer it over libcurl easy API.
I haven't noticed any memory leaks (server-side) with Poco 1.4.2p1 HTTPS server.
Another bonus, I've never seen guenter verbally assault anyone for asking a pretty well-crafted question/suggestion. I saw a newbie post a fairly reasonable suggestion on libcurl (here's a bug, perhaps you can fix it this way, not sure if it is the proper way to fix) and he was ripped to shreds by an angry badger attack. So if you go with libcurl, just make sure you don't use the word "perhaps" when posting a question -- and make sure you submit a tested patch instead of offering a suggestion. I looked for the post (Nov 2011 or Dec 2011) but was unable to find it because I don't remember the subject or author (the attacker appears too many posts to filter out.) I never felt so much happiness that my question was simply ignored and spared from such a response.
If you are new to C++, then I recommend C++ Primer 4th Edition (look for Barbara Moo as the author.) You picked a good time to learn C++. It is making a comeback and C++2011 and LLVM 3.x/clang will probably increase the popularity even more.