Yes indeed, you should style visited links. Just as you thought it would be useful for you, the same happens to your users. Of course this has been the object of many studies and research, so I won't extend on this, but the ability to tell where you have been is paramount in terms of Information Architecture.
Just a quick snippet from Nielsen about visited and non visited link styling:
Hypertext theory, the Web's history, and current design conventions
all indicate the need to change the color of visited links. Further,
empirical observations from user testing have identified several
severe usability problems on sites that violate this convention. When
sites use the same color for visited and unvisited links, users:
- unintentionally revisit the same pages repeatedly;
- get lost more easily because their understanding of each link's
meaning is reduced;
- often misinterpret or overlook the difference between two similar
links if they're unsure about which one they've already visited;
- and give up faster because they have a reduced sense of mastery when the
site fails to reflect their actions and thus help them navigate. Such
usability problems are particularly damaging to users with weak
short-term memory, who often have trouble remembering what they've
clicked without a visual representation.
Of course, "weak short-term
memory" is an inherent shortcoming of all humans, which is why all
users are harmed by unchanging link colors. But this definitely
impacts some people more than others, so it's particularly important
to change link colors if you have many older users.
To address the second part of your answer, would be quite opinion based, but given your links are black and your text is black as well, it would be advisable to style links differently. Keeping the same color for regular text and links is really confusing for users, so try alternatives based on branding or other elements of the page