Energy efficient bulb takes a long time to get bright
That type of bulb base is called a GU10 and should be widely available at home improvement stores, lighting stores, etc.

That particular bulb is a CFL (compact fluorescent) and the behavior you describe is pretty typical for these bulbs though some are better than others. An incandescent,  halogen or LED bulb will produce light at full intensity almost immediately.  Incandescent being the cheapest and LED the most expensive.