You can use the calibre companion app (paid but so worth it) by connecting a tablet either as a wireless device, which uses the new wireless device driver connectivity, or to the content server. The wireless device driver works as you'd expect any other device to work, just without wires. The content server connection lets you browse from your tablet and initiate transfers from there. Essentially, you would probably do the former for large numbers of books and the latter if you want to grab a few or a specific book from calibre.
From that point, in the Calibre Companion app, you select the book you want to read and it will ask you what app to use. You can set a default at that time if you want, and from then on it will always open that file type with the app you selected.
There are a few that do text to speech. Moon+ Pro does, and Cool Reader also. There are probably others but I haven't used them. Both of the ones I mentioned support epub and (non-DRM) mobi files, among other formats.
You can find more info on Calibre Companion in the Calibre > Devices forum.