A safer place for radioactives is one of the moons of Jupiter.
Thrusters are crucial to get more out of the fuel, which is important at the early stages when even fuel is expensive.
I agree that Alpha Centauri is not good for mining (although the surrounding stars may be), but visit anyway for a vital interaction as
Laiocfar said. A better bet I think is the Volantis system, it even throws in some bio-life for your hunting pleasure
. I never buy more than one lander in the game. Do not venture far from home until the ship is fully fitted with thrusters and turning jets at least. Do not leave home without visiting Pluto, it could well mean the difference between survival and a quick death.
As
Japofran said, talking to aliens and the starbase commander provides hints on what to do next. But when you're ready to start going farther out, you could follow the story arc and try heading to Vela. You could drop by Giclas/Krueger systems for some *ahem* freebies. But my preference would be to visit the Spathi, those in the know will know why :angel:.
An aside about mining. Once you feel reasonably powerful (weapon modules/escort ships) and comfortable in ship controls, forget about mining and mine your enemies instead. No cargo bays, higher and quicker returns but perhaps higher risk. The Mycon are particularly easy and profitable LOL.