Improvements like your Manufacturing Capital and Power Plants will boost production for factories on the planet where they are built by a percentage. The more factories, the greater effect they have. Research co-ordination centers and Technology capital boost the output of labs on the planets where they are built. Locating these improvements on specialized planets with a lot of factories or labs greatly enhances their value. Output from factories and labs can be further enhanced if they are inside the influence of an economy starbase.

During the colony rush, you want to find high quality planets - preferably with bonus tiles - to develop as your capitals. I typically build my economy capital on my homeworld - you'll be mid game before any other colony has a higher population. You need at least one decent research colony and several ship building planets but most of my colonies only get a market center until my economy has stabilized. Manufacturing worlds in the early game should be specialized - you want to build ships quickly. For research, I tend to build labs whereever I find bonus tiles and hold off establishing my technology capital until I find a high PQ planet - high teens or twenties.

Specialization is in my opinion the most efficient use of your planets.  I specialize all of mine for either tech, econ, or industry right from the beginning of the game.  This will increase your benefit from the structures like the various capitals, power plants, and research coordination centers.

As for what mix of specialized planets to use, it depends on your play style, and if you are playing for points or purely fun.

Sentient species taste better...