design involves quite a bit of trial and error, due to the large # of components that can be designed, and hence ships that can be made. On both the f2(economy) screen, and the f5(design ship) screen there are button labeled design and design tech respectively. Pressing either brings up the design component window. Provided you have the technology necessary you can build a component to your specifications, at which point that component appears in the appropriate tech list as a racial tech. At this point you need to have a researcher research said tech(or you can delete it if you want to redesign the tech). Then after it is researched you can design a ship with that component.
Once you have your ship designed find an appropriate shipyard, meaning the shipyard has capacity to build said ship, AND is able to build the design. Commercial yards can not build warships for instance. Naval yards can build all ships but are the most expensive to add tonnage to so it is generally best to limit such yards to small civilian scouts or military vessels. Once you have such a yard you have to retool the yard to build the design. If this is the first retooling that is free, otherwise you get to pay for, and wait for the yard to finish. Be sure you have the TNE resources required to retool the yard, duranium and neutronium in the desired quantities. This shouldn't be an issue for your first ship design but it will come up.
Once you have researched the right parts, designed the right components, expanded and retooled the appropriate shipyard you can build your ship, again provided you have the appropriate TNE resources to build said ship. After that it will be in the Shipyard TG(or another TG if you specified that) and you can move your ship to a different TG or just order it around as you want.