How to join miter cuts for picture/sign frame?

If you don't have access to a proper framing underpinner (friend's or neighbour's), then you could either ask a local picture framer to join them for you or if you do want to do it yourself, a pocket-hole jig might be suitable as this will join the corners from the inside where it won't show.

Always back up the join with a good wood glue too.

If you do go down the route of finding someone locally who make picture frames, their guillotine will also get rid of those gaps in the join.

Edit : details of how an underpinner works:

An underpinner is a framing tool that pushes small (maybe 5mm to a side, and ranging from 10mm-15mm deep), 'L'-shaped pieces of metal into the frame from the underside. This is the model I have :

a picture of the pins in place :

and here's a description of the process : http://blog.ukpictureframingsupplies.co.uk/picture-frame-joining/picture-frame-construction-corner-joining-with-an-underpinner/

I would recommend using a miter spline joint. There few 2 orientations to accomplish different looks. A quick Google image search will show you the final look.

For the gaps, they are not big thus wood filler and sanding will fix it. But if you really want perfect corners, I would recommend building a sled for your table saw. That is the most accurate way to cut a 45 angle.