A software that includes AGPLv3-licensed components is subject to the AGPLv3 as a whole. But other parts of the software can be under compatible licenses such as the Apache 2.0 license.
So yes, you can combine AGPLv3 and Apache 2.0 libraries, with the resulting software being subject to AGPLv3 as well.
Roughly, AGPLv3-licensed software implies the following obligations:
- you are not required to publish your software. It is fine to keep it private.
- the AGPLv3 is almost identical to the GPLv3
- you must include a copy of the license
- modified versions must have the same license
- if you give someone a copy of the software, you must give them the source code under the same license
- additionally, if you modify the software and let users interact with the software over a network, then the software must offer users the source code for your modified version.
All of this only applies if you want to give someone else a copy of the software, or want to make the software available over a network. Purely internal use (e.g. personal use or within a research group) does not trigger any license obligations.