Follow the NOOBS Setup instructions.
Update Raspbian from its Debian wheezy base to Debian jessie:
sudo vi /etc/apt/sources.list # or use your favourite editor
change all references of wheezy to jessie
sudo apt-get update
sudo apt-get upgrade # this will take a long time, with occasional user prompts
sudo apt-get dist-...
Based on available documentation, it sounds like you need to :
Install a webserver on the Pi, like Apache
Place the tiles (which you said you've already generated) into a known directory structure within your web server.
Per the previously link: Change the OpenLayers instance to use your own tileserver instead of the main one
The link in the 2nd point ...
I believe you're trying to ask too much out of that hardware... running a GeoServer requires a "server" (a laptop PC will do, even a few years old), the Raspberry does not have enough computational power.
I believe that running a full fledged distro like Ubuntu MATE is not helping, but the real issue is the hardware.
According to http://qgis.org/debian-nightly/dists/jessie/main/, the QGIS binaries are for amd64 and i386 only. If you look into How to install GDAL and QGIS on a Raspberry Pi? you will learn that Raspbian needs armhf packages, which are not hosted at qgis.org. This is basically what the last error message tells you too.
So you have to use the old packages ...
From what you've described, a simple option would be to use Tilestache. It supports mbtiles natively, so you don't need to unpack them. That is good for maintenance purposes, and can save space because of the view idea I described in a comment on another of your questions.
There are at least three ways to serve tiles with Tilestache. I'd go with the first ...
Alternative to building from source.
Issue: with the QGIS install instructions (assuming I followed them correctly) - it was unable to find the python3-qgis package.
Solution: basically use a different mirror than what the install recommends.
Starting from a fresh Raspbian Stretch with Desktop install.
Run: sudo apt-get update
Run: sudo nano /etc/apt/...
openmaptiles.org offers mbtiles downloads for the whole planet as well as for individual countries and cities. It is the successor of osm2vectortiles.org.
It contains also a documentation for generating your own OpenMapTiles.
I've had not unreasonable results with my RaspberryPi 3, though I'm not trying to do any complex rendering yet. You can see a lag when panning or scrolling a map but using tiles and caching helps with that (as you'd expect).
I wouldn't want to use it as a server for production but as a test platform and for taking out in the field with no power it's fine. ...