What do I need for a good flow of water from a faucet: velocity of water or pressure?

I can't understand one thing. A good flow of water from bath faucet requires a good velocity of water or a good pressure? For example, I have a crane that has a pipe diameter of 1 inch. The better the flow of water, the higher velocity of water. But the higher velocity of water, the lower pressure (bernoulli principle). So it seems that the flow is good, but the pressure is low.

I think the key misconception here is that, in case of flow through a pipe (which is what feeds the faucet), Bernoulli's Principle is not applicable. If you read the Wikipedia page, one of the assumptions for Bernoulli's momentum equation to hold is that the flow is inviscid, i.e. the effects of friction are negligible. However, in flow through a pipe, the effect of friction is what dominates, since friction is the main force that is opposing the flow, as it moves.

When water exits a faucet, the jet almost instantaneously equalizes to atmospheric pressure (since the jet is free to expand/contract). For flow through a pipe, the Darcy-Weisbach equation should be used, which states that the head loss (i.e. pressure loss, $\Delta p$) through a pipe is proportional to $v^2$. So, in order to get a higher-velocity flow through a pipe, you need to have a bigger pressure drop through the pipe - between the start of the pipe and where the pipe/faucet is discharging. Essentially, this means having a bigger pump at the start of the pipe.

The flow rate can also be increased by removing other sources of 'head loss', such as opening a valve, or using a less-restrictive faucet (or shower head).

When the water is coming out of a faucet what matters and can be measured is its velocity and the flow rate. Ideally, both should be high, in which case it would obviously be a good flow, but it is not always achievable.

Given a certain static pressure, the flow rate would depend on the total resistance in the pipes, valves, etc. Naturally, a high static pressure and a low resistance will produce high flow rate.

Since the static pressure is likely to depend on external factors, all we can do to increase the flow rate is try to reduce the resistance, which may involve replacing or cleaning an aerator, a shower head, a flow restrictor, etc.

If the flow rate is OK, but you wish to increase the velocity, it could, in principle, be achieved by reducing the size of the faucet opening. We are all familiar with this effect: you block part of a faucet with your finger or squeeze the end of a hose and the velocity of the stream sharply increases.

This restriction of the flow at the exit has a very different effect than the restriction of the flow by a valve, which just reduces the flow rate. This is because, the kinetic energy of a high speed stream coming out of the valve is quickly lost on its way to the exit of the faucet, which is not the case with the restriction at the exit, with no more obstacles in front of the steam.