Although four layers of copper does not sound as though it can make for many ways of implementing a board, the reality is rather different. And the results are not all the same, even if the electrical connectivity and hole and via positions are the same. If the materials are not specified, the separation between layers not stated, and the layer sequence are not provided to the company that makes the bare PCB, you could get almost anything when you receive your boards. It can be enough to make the difference between passing or failing EMC tests. The decisions made are therefore important.
This does not mean that there is only one right answer. There can be quite a few right answers though not all will be good answers. As with many things, it is a question of choosing a good balance.
Suppose you want your board to have USB2 or USB3, these expect a differential impedance of 90 Ohms or 85 Ohms. But you have lots of tracks on the board so you need these differential pairs to be fairly narrow traces with say a 2x trace width gap between them. To do this you need a thin gap (prepreg) between the outer layer (where your signals are) and the reference plane immediately beneath. This favours a thick core at the centre and then a thin prepreg and foil attached to create the top and bottom layers.
If you want some really high frequency differential pairs, one of your main concerns is signal loss. To reduce the loss you need wider traces, but wider traces need a bigger gap between signal layer and plane, so you shift the balance of thicknesses to make the prepreg be thicker (or just use two or three of them).
In general, cores can be very thin, but you can get a whole range of thicknesses up to the thickness of your whole board. Prepregs are always thin. The copper layers can also vary in thickness, 18um and 35um being two of the most common sizes. You could find the data sheets for the PCB materials and work out what combination works best for you; but the better way is to talk to you PCB supplier. If you ask them for their advice on cores and prepregs, and you tell them the things that matter in your design (like perhaps controlled impedance) they will probably suggest a good combination of materials to design with - but better, they’ll do this knowing that they are recommending materials they have in stock and that will probably give you a good price with it.
The last thing about choosing a stack-up is choosing what to put on each layer. With four layers it is tempting to think that the layer allocation should simply be Signal, Power, Ground, Signal; but if you’re using frequencies over say 10 MHz, this stack-up is a poor choice. A much better one is Signal & Power, Ground, Ground, Signal & Power. This is good for medium/high/very high frequency work and done well is excellent for EMI (if you remember the rule that where you put in a via to move you signal between top and bottom layers, add a Ground to Ground via close by. Route the power as a thin signal to begin with, then thicken it using polygon pours. This approach yields surprisingly good performance, and is easy to implement. From experience, this method allows USB2/3/4, PCIe, HDMI to be tracked on 4-layer boards that are regularly 15dB clear of the class B limit lines for radiated emissions, even on peak detect mode (and better on quasi-peak measurements). On the other hand, getting close to this with Signal, Power, Ground, Signal stackup is impossible (or more accurately, I’ve never seen it done).
Key take-away: not all 4-layer boards are created equal, and the differences can be large. For the fine detail, talk to the PCB manufacturing company you want to use and take their advice. Give them more detail in what you’re looking for and you should get a good lead on what you need. Once you’ve got some sort of idea about what you want, look at the Stack-up designer tool on the Sierra website. There are other tools on this site that are available to use as well.
Hope that helps a little.