How to serve tiles from my own server?

To get your own map tile server, you want OSM in a Box - it does all the hard integration work for you :

• Configures PostgreSQL/PostGIS server
• Creates an empty database with geospatial features - keeps all attributes in hstore
• Configures GeoServer and tries to keep osm2gis-configuration consistent with GeoServer styling rules
• Loads OSM data into the database
• Incrementally updates the database
• Produces nice maps (tiles and cached through GeoWebCache)

You can start with the OpenLayers simple example.

Note the section for "Other tile sets". You have to add this code to the layer definition section, instead of the two lines for Mapnik:

var newLayer = new OpenLayers.Layer.OSM("New Layer", "URL_TO_TILES/${z}/${x}/${y}.png", {numZoomLevels: 19});
map.addLayer(newLayer);

Just change the URL to the directory where your tiles are, and the zoom levels to how many you have. Then open it in your browser, and it should be showing your local tiles.