Why would you want a trebuchet? I know quite a few reasons since I have already build two of them. Most likely you would like to have a one because it is a really cool prop and a great demonstration of simple applied physics. Any other reasons are of course valid points to start building.
Why would one want to build two of them, you might ask. The main reason for the second one was portability. I wanted to transport my trebuchet by car without problems, which is nearly impossible with reasonably sized one. On the other hand, I wanted to have an option of setting it up without any tools in a reasonably small amount of time. This was a challenge of its own, therefore the other aspect, performance, wasn't valued very highly during the design process. Portability and performance require quite a different set of attributes in the final product and they do not like each other that much.
I decided to go with the classical design with fixed main axle. There are many designs with floating axle, the most prominent one being Merlin trebuchet. These do not resemble the original medieval design in any aspect, on the other hand, if you would like to get the maximum distance for set counterweight, this is the one you want.
Dimensions were partially set by the available materials, partially what has proven over time on the first build I did a few years ago. Measurements of the important parts are shown in a diagram below.
Considering the measurements, there is only one part you should be concerned about, the main swing arm. All other components can be built around it just to fit, regardless of the exact dimensions. The magic number, you are looking 3.75:1. This is the ratio of swing arm proportions, i.e. the long part of the swing arm is 3.75 longer than the short part. The sling should have the same length as the longer part of the swing arm. However, the ratio is not set in stone. There are trebuchets that have ratio 5:1 or even 6:1, as well as the other part of the spectrum, the ratio can be 2,5:1. The main point here is following, there has to be enough leverage to make the swing fast enough. Therefore larger trebuchet with heavier counterweight can make it work with a more extreme ratio, while small desktop model will struggle to work with the optimal 3.75:1 ratio. It is all about making it work best for your setup.
The second most important measurement is also a ratio, this time counterweight mass compared to projectile mass. Ideally, this ratio should be around 133:1, but same as before, your settings might vary and having some extra mass doesn't hurt. Therefore if you just want working trebuchet, stick with the recommended values. For all the others, take the recommended values as a starting point and run some tests for your setup, maybe even simulations, as there are online solutions available for testing dimensions and weights.
Building a usual trebuchet, i.e. permanent one, is rather simple and not much-complicated thing to do. Select your material - the best option is usually wood, since it is easy to build with, does not weigh too much and can withstand reasonable forces. This is the most important part, you should have reasonable expectations how much counterweight you are going to use and select your material according to that. This particular one is mostly limited by the portability and individual parts not being as strongly connected together as they could be in permanent one. Otherwise, this selection of material (construction wood, individual pieces approximately 4x6 cm wide) could withstand about 100 Kg of counterweight, provided right selection of joinery and screws.
The main fulcrum (axle) should be made out of steel rod and the diameter should again correspond to the expected counterweight. Rod with 16 mm in diameter should not carry more than 15 Kg, 25 mm or even 30 mm should be good for 100 Kg. This is by no means accurate, just from my experience. For the most simple design, just straight hole through the wood will be enough, maybe consider some hardwood for the mounts, as it will last longer. For better performance, you can include bearings or steel inserts into the build. The secondary fulcrum usually does not require any special treatment, since it has very little effect on the overall performance. Also, the steel rod can be of smaller diameter, since it is usually not prone to bending.
All photos are accesible in gallery here.