You can use both LGPL and AGPL code, provided that you link to them. The GPL and AGPL both include exceptions allowing linking to code under the other:
Notwithstanding any other provision of this License, you have permission to link or combine any covered work with a work licensed under version 3 of the GNU Affero General Public License into a single combined work, and to convey the resulting work. The terms of this License will continue to apply to the part which is the covered work, but the special requirements of the GNU Affero General Public License, section 13, concerning interaction through a network will apply to the combination as such.
and the LGPL specifically aims to allow linking. As long as you're not copying code (and assuming all the projects can be distributed under (A/L)GPLv3), the project as a whole will be under GPLv3.
See this table for more information.