There are different types of mirrorless cameras. There are APS-C mirrorless cameras like the Sony NEX series and Pentax K01 which have every bit the image quality of an APS-C DSLR. They’re not as compact because they do use the bigger DSLR lenses.
Most of the mirrorless cameras which use four thirds or micro four thirds, which is a significantly smaller sensor size than APS-C. The benefit of these systems is that they were designed to be mirrorless from the ground up - their decisions about lens mount design allow lenses to be smaller and to function better for contrast detect AF - so they become smaller overall systems, but their image quality lacks compared to APS-C.
With advances in making APS-C cameras smaller, I’m not sure how much of a future there is in 4/3rds. I wouldn’t want to invest in it right now. I’d put my Pentax K5 with the small limited edition primes against any 4/3rds system and beat them in size and image quality - but if I wanted a big telephoto, it’d be noticibly bigger than the 4/3rds equivelant.
The NEX-7 is an odd duck ergonomically but it’s as good as any APS-C camera can be in terms of image quality. But the Olympus OMD-M5 is the new hotness in 4/3rds. A retro rangefinder style body with a pretty good sensor, that’s a tempting way to go.
The newest mirrorless cameras have a cool feature called focus peaking, where it highlights the areas with the most focus (similar to a zebra display for blown highlights) which actually make them more effective combined with manual focus. Ironically, they are usually not made in the mounts that can use all the old high quality manual glass though. That feature will probably make it into DSLR live view modes though.
I’ve never used an electronic viewfinder, but a lot of people can’t stand them. There’s always a slight lag time, they’re never quite really the real image, they’re not as easy to track with - but some of the newer ones, especially the sony ones are their high end cameras, are very well regarded. And they can do stuff that optical viewfinders can’t, like zooming for focus, showing changes of the effect of exposures while you dial it in, reflect any specialty modes like black and white set in the camera, etc.
Personally, I figure… mirrorless cameras are never going to fit in your pocket, so they’re not going to be a camera you can always have on you. So you’ll only take them when you specifically forsee photograph opportunities. But if you’re going to be dragging along photographic equipment anyway, you might as well go a little bigger an get a DSLR. Not all DSLRs are giant Nikon monsters - the aforementioned K5 isn’t much bigger than most mirrorless cameras, but has the best sensor of any APS-C camera out there.