Welcome to the forum!
Safety concerns:
- always use food-safe containers and materials.
- always check to make sure that something you're planning to use/add is safe for your plants and your fish and you. (A lot of common garden chemicals aren't safe for fish; a lot of common aquarium products aren't safe to use on anything you plan to eat.)
Common mistakes:
- failure to plan for power cuts (i.e. tank will drain if pump isn't going, fish are stocked at a level where they'll die in hours if the air pump shuts off and there's no backup).
- stocking way too many fish because they're tiny now and surely they won't produce too much waste for your system bacteria to handle (yes. yes they will. baby fish are growing fast and produce loads of waste.)
Good but inexpensive materials:
- IBCs and food storage barrels! Black or dark blue barrels last best in sunlight; IBC bladders will last several years if unmodified and even longer if painted / wrapped / cladded in some way. (Plus this prevents algae growth.)
- Second-hand things, for example there are some awesome system threads on this forum with fish tanks made out of cut-down water storage tanks and/or growbeds made out of recycled bathtubs.
- Home-built stuff if you have the skills and time! People have built all sorts of things out of concrete or blocks, welded frames with timber and pond liner, you name it.
Different methods:
- This one varies wildly depending on how much space you have, how big a system you want to build, how much you like tinkering versus automation versus complexity versus simplicity, how much time you're willing to spend on regular maintenance and what / how much you want to grow. The simplest hobby/backyard systems are usually mostly media growbeds, because they combine planting space, mechanical filtration, and chemical filtration all in one. Commercial and intensive systems seem to lean towards purpose-built filters and lots of DWC. If you want to grow lots of leafy greens and are going to be taking several whole plants out every time you harvest, DWC works well for that; if you want to grow large perennial/fruiting plants that will be staying in the system for months or years and are likely to develop huge root systems, media beds might be better for you. NFT seems to work a lot better if you don't have extreme temperatures or are doing something to compensate for that.
Your best bet is probably to read lots of system threads to see what other people have done and how it worked out for them, and also stuff like the IBC of Aquaponics (which is linked up on the very top of every page on this forum, in the thin black bar above the header). Given that you're trying to get your family interested and for now at least you're probably the person most willing to actually work at this, I suggest building something fairly small that you can take care of yourself until you win them all over with the awesomeness of your produce!
A single chop-and-flip IBC or something built using half barrels is a good start, is very flexible and small enough to be manageable while you're learning while still big enough to be fairly stable (temperature and water chemistry), and is expandable once you get some more family members bitten by the aquaponics bug. 
I built a tiny system on the back verandah out of tubs I bought from the local hardware store, to make sure that I was interested enough to stick with it if I went bigger (and win my husband over to the idea with tasty veggies); now I'm running a pretty large system and expanding my growbeds!