Everything in OGame comes from three resources: metal, crystal and deuterium. Early on, your absolute priority is the economy: a solid mine base funds fleet, defense and research. Players who neglect mines stall.
The recommended order
Raise the Metal Mine and Crystal Mine together, and never forget the Solar Plant: without enough energy, mines produce below capacity. Deuterium becomes important a bit later, once you start researching and moving fleet.
A practical rule for a new miner: keep the crystal mine a couple of levels below the metal mine, since crystal is pricier to upgrade but more valuable for research and ships.
Do not neglect energy
Each mine level consumes more energy. If your energy balance goes negative, production drops proportionally. Alternate mine upgrades with the Solar Plant, and later weigh the Fusion Reactor or Solar Satellites depending on your planet temperature.
Up to what level?
The cost of each level grows exponentially, so there is a point where upgrading a mine takes too long to "pay off". The decision depends on your universe speed and whether you prefer to grow as a miner or reinvest in fleet.
Plan with numbers
Use the Costs Calculator to see the cumulative cost, the real build time (based on your Robotics and Nanite factories) and the points you earn per level. That way you decide with data instead of guessing.