I think the earlier one starts doing research the better, even if it is a research in plane geometry.
I directed a few REU's and it was great fun; the only problem is that it is hard to do anything significant in 8 weeks. I am confident that many US undergrads can produce a publishable work after focusing on a problem for 1-2 years.
In the place where I went in college (Novosibirk, Russia, mid 80s) students went through abstract algebra and real analysis in the first two years and the best of them them could do nontrivial work in the 3rd year. Quite a few people were going to research seminars in 3-4th year, which they had to be doing since a (master) thesis with original research was expected at the end of 5th year. As it happens for many students this thesis was largely expository, but those who later become professional mathematicians usually got something publishable in the 5th year. My own research in the 4th year went nowhere, but in the 5th year I proved something I am not ashamed of.