What kind of cache-related attacks exists in modern Web world? The only one I'm aware of is Cache Poisoning. Is there anything else I should be worrying about?
There is a lot going on in the infrastructure that you shouldn't be worried about, so long as you keep these systems up to date. Browsers, proxies and other systems have gotten really smart over the years and if you asked this same question 10 years ago I would have given you a completely different answer.
DNS Cache Poisoning is a very serious vulnerability, but so many of these have been found that its really difficult to preform this attack on a modern DNS server.
DNS Cache poisoning has a very similar impact to ARP table poisoning. At the end of the day a table is just a type of cache.
Did you know that web browsers will encrypt cached copies of all HTTPS content? That is really nice. HTTP proxies are unable to cache this content so that is a good thing as long as you use https. However the cache used by HTTP proxies can be poisoned using HTTP Response Splitting which can be a serious problem.