[henrik_w]

Yes, but I have to hit enter first. And, having hit n a couple of times, I haven't found a way to easily add more letters to the search pattern (but I am pretty new to Vim). Also, I (believe) I need to escape out of search mode if I want to use * if I found the word I was looking for and want to find more occurrences of it.

[johncoltrane]

You can use <C-r><C-w> to insert the word under the cursor in the command-line. You can use <C-r>/ t oinsert the latest search pattern. You can use <C-w> to delete the word on the left or <C-u> to delete everything you already entered. You can use // and ?? to reuse the latest pattern.