In order to research new technology and componenets for your metal maidens, you need to spend codon and research materials at the science and technology area. You must spend codon to research new stages of each tech tree, and must use codon and materials to purchase the technologies. Once a technology has been researched, you must spend silver to mount it on a tank.
Tanks can't equip all technologies, make sure your tanks can equip what you are researching
Technologies have certain prerequisites before you can research them - Your commander level needs to be high enough and you need to have researched the previous techs. Tapping on a tech will give more info.