Rockets come in all shapes and sizes; but why not start at a tiny size - using things that you already have or can buy with loose change.
Alka Seltzer Rocket
The Alka Seltzer rocket rocket is a classic science experiment. You put an Alka Selzter (or equivalent) into a container with some water, and once the gas builds up it pops open to launch the rocket. There's no shortage of information on the Internet about how to build them - this is a good example - but almost all use film canisters. You can still buy them, and if you are working with a group then you can get good deals on bulk orders from ebay; but they aren't the sort of thing that most people have lying around.
You might have an alternative lying around, but you might want to test them first just to be sure. A likely candidate is the inside of a Kinder Surprise, but alas the two-part canisters of childhood have been 'improved'. The image below shows the break in the rim (near the joint), that allows both gas and liquid to squirt out of the side in a very un-rocket-like manner. A better alternative is the glove container that comes with some hair dye products. Although you're only slightly more likely to have one of these than you would a film canister.
When it comes to building the rocket itself you don't need to be too precise - it's just a paper tube for the body, a circle with a slit - folded into a cone - and some triangles for fins. Just use sticky tape to hold it all together.
I once saw a presentation by Jerry Stone, and his top-tip was to secure the tablet in the smaller piece (the lid of film canister in his case) instead of just throwing it in and rushing to connect the parts together. I think he used blue-tack, but a loop of sticky tape works as well.
Feel free to experiment with different 'fuels' and 'fuel-mixes'. I had just as good results using some vitamin C tablets as with the Alka Setltzer. Pretty much anything described as 'effervescing' will work. Using just enough water to cover the tablet also seems to work best. There's probably some science behind that related to the compression of gases and liquids.
When you light a match you see an initial burst of flame as the chemicals in the match head ignite. As with most combustion this produces a certain amount of gases - which normally just escape into the atmosphere. By wrapping the match head in foil you can funnel these gases in one direction - like a rocket engine. It's all summed up in a video by MarekSHayward:
The matchbox launcher is a very good idea; but maybe you want to keep your matchbox for storing matches? There was also an opportunity to make some design improvements - a longer drawer for added safety, and a grip for easier handling. You can download the template to build your own below:
Once assembled, you'll end-up with something that looks like this: