I've never tried to program with a visual interface like this; just Python and BASIC and stuff. I just need output 1 to be on for 10 seconds, then shut off, have output two be on for 5 seconds, then shut off, and repeat for infinity.

Surely there a way to do this with LOGO! or just a written language I can use to program a 0BA8?

I know I could do this with a timer relay but I'd rather just use the PLC because reasons. 

Wow, I've jsut now seen that your question remained unanswered for a long time.

Sure, it's possible and it's one of the things that a small PLC is made for.

First you have to let go of a few Ideas from procedural programming, a PLC works differently.

A PLC works in a cycle. At the start of the cycle it reads all inputs, than it works trough the whole program and at the end all outputs are set after which it starts a new cycle.

So, you don't need loops and branches are also not really a thing.

What you want is not hard to do. If I understand you correctly, you basically want to switch between two outputs, so when Q1 is on, Q2 is off and vice versa.

You just use an Asynchronous Pulse Generator and an inverter and two output blocks. Invert the input of the Asynchronous pulse generator (so it always runs), set the times according to your needs in the pulse generator, connect the output to an Output block and an inverter and connect the output of that to the second Output block (see attached image)

Oh, a really smart solution with the triggering on the negative edge... Opens the possibility for more stages really easily.

And with 2 additional flag blocks and an or one could even eliminate the 1 cycle break when none of the outputs are "on" in this solution...