The free version of Malwarebytes is a 'run on demand' program. (i.e. it does absolutely nothing unless you actively run a scan with it). So it can't conflict with any other security software and, as I'd regard it as essential on any internet-connected computer, you'd be mad to get rid of it. (Remember to run it periodically though; perhaps once per fortnight)
However having two anti-virus programs 'running in the background' simultaneously is asking for trouble. You definitely shouldn't have McAfee and MSE running together! Since any 'free' version of McAfee will be time-limited (and will then ask you for payment in order to keep using it), that's what I'd get rid of. The instructions for completely getting rid of it are here
https://service.mcafee.com/FAQDocument.aspx?id=TS101331
They make it seem incredibly complicated but all it actually says is:
1. Go to your list of installed programs and uninstall ' McAfee SecurityCenter' ; then
2. Download and run this file:
http://us.mcafee.com/apps/supporttools/mcpr/mcpr.asp
(In practice you can probably get away without running Step 2 anyway)
Once you've got rid of McAfee, you can either continue using MSE or uninstall it and then install Avast (as above).