Yes, you are obliged to make the source code available, under the same terms as the AGPL - which are almost equivalent to the GPL, by design.
Technically, the GPL requires that source is distributed only to people who ask for it, and who have received a copy of the binary. You are not obliged to go beyond that, although you cant stop, eg, me from giving the binary to Mary, and Mary asking for the source through me.
The additional nature of the AGPL is that if you put the AGPL component into a service that is never distributed, but is used by others, you are also obliged to make it available to them.
Since the AGPLv3 and the GPLv3 are almost identical, "Does the GPL allow me to sell copies of the program for money?" applies to you also. In other words, "yes, totally, go right ahead".
You can't stop me buying a copy, getting the source, and distributing both free of charge - but you can totally put it, eg, in the iOS market and charge anything you want. (You can't stop me doing the same, and pricing it one cent lower than you do, though, so the model can be ... risky.)