He asked me for some to bring blood (not mine). [I don't know how to insert a spoiler so I'll try to be vague.] On the way out of his den, I met another character whose offer I refused. Now I don't know where to go or what to do. There are no guiding arrows directing me to a specific place or pointing to a destination on the map. I can't just go around spilling other people's blood just to satisfy Septimus, even if I thought it would do any good. Should I have accepted the offer of the character at the exit? I suspect that was my error.

just find some Mer bandits, kill them and take their blood to Septimus

If you go back to the place you were just at where you went into Black Reach you can get all the elf blood except Dunmer, which you can usually find a Necromancer at the ritual stone. Other than that you can just go dungeon crawling until you find bandits that fit the races you need.

Edit for future reference you should probably use the same topic you already made on a subject and if it has spoilers to put it in the spoiler section of the forum.

All the blood is within Whiterun Hold. Dunmer at Drelas Cottage, Orc and Bosmer at bandit camps, Falmer at Shimmermist Cave, dead Altmer at exit of Halted Steam Camp.

There is so much to learn about using the forums. I notice some people hide spoilers in little boxes in their posts, but I can't figure out how to do that.

Type [spoiler], whatever the spoiler is, then close by typing [/ spoiler] but without the space.

Use the Special BB Code button - third from left, top row.

Posting in the Hints & Spoilers forum is also recommended

At least you can admit it, you try to help some people with the forums and they tear you head off.