This was the textbook for the first third of a year-long algebraic topology sequence at Oregon State in 1973-4. We were told by the prof that Vick was a student of Stong and that the book was essentially Stong's course written up with his blessing. It's hard to ask for a better pedigree than that, as Stong was a legend for his teaching (as well as his research).
Although there are some minor quibbles (noted in the 4-star reviews), I still haven't found a better treatment of the key results, nor a more direct path. The proof of Poincare duality in particular is that of Hans Samelson, another legend in the field for both teaching and research.
Checking the references, one finds that this was not the only such example where Vick sought out what was then regarded as the best proof available for beginners. It is also noteworthy that community consensus on which are best has not changed much, if any, since then.
The plethora of typos may be a "feature" of the reprint, since I don't recall that many in the original Acad. Press edition we used, and I still have.
As should be clear, this one is a real keeper.
For more modern/advanced study, continue with Switzer and Brayton Gray. By then the journals should be reasonably accessible.