Would a spaceship have to evade interstellar clouds? Apparently, there are clouds of "dust" between the stars. Would a starship have to fly around those clouds, trying to find "tunnels" between clouds, or are the interstellar clouds harmless for a spaceship?

I'm mainly thinking in terms of abrasion or (micro)collisions, not radiation, but would welcome information on the latter also.
Yes.

As has been commented, the amount of damage taken by an interstellar spaceship depends on its velocity $v$, as well as the number of gas and dust particles that it encounters on its way. This number is usually measured per area, in which case it's called column density $N$, and is equal to the total distance $d$ traveled times the particle density $n$, i.e. $N = nd$. For instance, if a spaceship travels 1 lightyear ($10^{18}\,\mathrm{cm}$) through a region with a density of $10\,\mathrm{cm}^{-3}\!$, each square centimeter of the spaceship will encounter $10^{19}$ particles.

That is, the faster you go, the farther you go, and the more dense regions you go through, the more your spaceship is damaged.

The Breakthrough Starshot project aims to reach our nearest-neighbor stellar system $\alpha$ Centauri in ~20 years, with a gram-sized satellite reaching $0.20c$ by means of a light sail. Today, there was a paper by Hoang et al. calculating the amount of damage taken by such a satellite. The total column density of gas from Earth to $\alpha$ Cen is $\sim10^{17.5\mathrm{-}18}\mathrm{cm}^{-2}$, and assuming (fairly) a dust-to-gas ratio of 1% and a carbonaceous/silicate dust grain population with a Weingartner &amp; Draine (2001) size distribution, they calculate that this journey to $\alpha$ Cen will erode the surface of the spacecraft to a thickness of the order of 1 mm.

Most of the damage is caused by dust, not gas, but in principle gas may slowly heat up the spaceship. However, at $v=0.2c$, as long as the density is $\lesssim10\,\mathrm{cm}^{-3}$, the temperature is insufficient to cause any melting.

Molecular clouds — the dense clouds where stars are born — have densities from $10^2\,\mathrm{cm}^{-3}$ and even up to $10^6\,\mathrm{cm}^{-3}$, i.e. many orders of magnitude higher than the roughly $1\,\mathrm{cm}^{-3}$ found in the more dilute interstellar medium. To reach even more distant stars in a tolerable time, you would have to go faster than $0.2c$, and thus it seems that it is in fact a good idea to evade these clouds.