Each race has a colony frigate that can be built right at the start from the frigate shipyard, which you can use to colonize asteroids, terran, and desert planets right away. Ice and Volcanics need additional research before you can colonize them. Additionally, each race has a capital ship capable of colonizing (Akkan for TEC, Jarassul Evacuator for Vasari, Progenitor Mothership for Advent).

Each planet, when colonized, produces its own construction frigates that get to build all the structures in their gravity wells, the construction ships themselves can't travel between planets. You never need to (but you can, if you wish) to select the construction ships to build, everything is accessible from selecting the planet - all the build options (ships, structures, planet upgrades) are there. Ore and crystal extractors specifically are in the Logistics menu (looks like a gear).

Some planetoids (gas giants, asteroid belts, junks, etc) can't be colonized but may contain neutral resource extractors that you can use colony frigates (TEC, Advent) or Scout ship (Vasari) to capture for your glorious empire

Hope this helps! Any more questions, feel free to ask