What is wrong if my espresso pours out slowly but the first drops shoot out? I've recently bought my first espresso machine (a Rancilio Silvia). I'm now trying to dial it in, but don't know how to interpret the result:

My shot is not too fast (it does not splash out of the machine). It also takes about 5 seconds until coffee starts pouring out. However, the coffee does not flow as evenly as I was expecting. The first drops really shoot out of the machine and then turn into a quivering stream of coffee. In the end, the cup is not clean at the rim but rather a bit sprinkled with coffee drops.

Did I tamp too hard or why does this happen?
Looks like you've got too much ground coffee in the basket. What's the weight that you're using? It looks like the recommended dose for the stock basket is 14g, which is a bit lower than the "standard" of 19g or so.

You have an indentation (ring) around the outside of the grounds -- an indentation from the group/dispersion screen (in addition to the notorious protruding center screw on this model). If you're already tamping thoroughly, this could mean you've got too much coffee in there.

Here's a thread from Home-Barista on the Silvia around a similar topic. It discusses another way of diagnosing the dose -- inserting the PF with tamped grounds, then removing while still dry. 

That article also talks about aiming for a dose+tamp to leave a bit of space between the tamped grounds and the screen: approximately the thickness of a 5-cent piece of space (1-2mm), which is a general recommendation (not just for this machine), though there is certainly disagreement on this. Another article from Serious Eats discusses this topic and notes that a "sludgy mess" can result, because the grounds will expand a bit when they absorb water. Your picture also looks a little "wet"; it could just be the picture, or perhaps a similar symptom.

Try a lower dose (14g) and do the dry-removal test, aiming for a bit of space between the top of the tamped grounds and the screen.