Is it okay to include other open-source packages in my own software package? In particular, I want to distribute my software (using Apache 2.0 license), and I would like to include another package, which uses the same license (my software compiles and links against it).
Also, I am using two boost libraries. Can I just include them in my tarball so that my users don't have to download them themselves?
As you have said how much background you have about licenses, I take the liberty for some general talk: In general, the terms of most software licenses vary upon if you want to distribute the source code; modify the source code and redistribute the source code; or whether you want to distribute a "derived work" in binary format. The point of misunderstanding/contention is often what constitutes "derived work".
The answer to most if not all of your questions is: it is ok for the licenses you mention -- Boost Software License (check this) and Apache 2.0 License (check this). Apache needs you to attribute their library along with your binary distribution and this is in general a good idea, IMO. This is often done by including a LICENSES.txt file containing all the third-party licenses and showing it to the users during the distribution.
But note: I'm a programmer, not a lawyer, so this can hardly compete for a legal advice. And I understand that the answer might be vague -- that's a limitation, not an intention. That does not mean I won't tell you what I know.
PS: Also I'm not sure if you want to release the source code or the binary for commercial use. In the case of binary for commercial use, it probably would be advisable to release your binary under a EULA rather than under Apache License.