In the ship construction window:
Click 'new' on the bottom. Go to the 'hulls' tab and select one of the available hulls. Start with 'tiny', as there won't be alot of room so it'll be a simple ship.
The next tab is the jewelry. Stuff to make the ships your own style. You can skip that for this...
The next tab is engines. If you want them, select an engine. It will list types and sizes depending on your tech level. Early on there are few choices, later you can have smaller, compact engines with decent speed. I personally don't use them, except for a specific class of ship to 'intercept' attacks... creatively called the 'interceptor' class.
Next up is weapons. Slect a weapon type, which again will be based on your tech level. Each component has a size, and each hull has a size limit. Your Tiny hull will have maybe 20 spaces, a medium hull maybe 70. This will change based on your races abilities and your miniturization tech level. With all the techs and buildings you can have 110+ spaces on a medium sized hull... That's alot of weapons room. 
On our first Tiny hull, you may have room for only one weapon. On a small hull, which I tend to use as heavy fighters you can have 2 to 6 weapons again depending on tech levels. Pick a weapon on the right, and then place it on the ship in the left view window. There are red 'hardpoints' on the ship that are used to attach thhings, be they weapons, jewelry or whatever. If you don't see them, look in the upper left corner of the ship view screen where the directional block is. The lower left has a circle or dot in it. Click it and it will toggle the red hardpoints on and off.
The next tab is defenses, and the last is modules, and they pretty much follow the same rules as the weapons.
In the setup or options screen, you can toggle 'auto-design ships for player' which will build you ships with components attached. It has a governor that will let you set priorities for types of weapons and defenses. You can also toggle 'remove components when upgrading' too.
To upgrade a ship, when first going into the ship window, highlite a ship and select 'upgrade' instead of new.
This is all NOT to be confused with the Upgrade command in the game screen when you highlite a ship. That's different. To use that, select upgrade. You'll get a list of all the possible ship configurations and their respective costs. Highlite the one you'd like and it will upgrade the ship to that type. You can also check the box to upgrade all the ships of that type.
If you have TA you can use the ship constructor from the main start menu without starting the game. You don't have to worry about tech restrictions, but remember the ships won't show up in game unless all the tech requirements for the design are present.
I like to design ships 'naked' and name them as ship classes. They are always there, and I just have to add the weapons/shields etc I want for any particular game. I also have all my ship designs available every game, so I can always use my favorites.
Hopefully helpful. I'll try to check back if you need anything else.
T
