I use the 5 speed Harbor Freight lathe. Today its fine for me. When I bought it, it wasn't. In a while (months, years, ???) it probably won't be. Here are my thoughts:
Everyone knows about the general quality at Harbor Freight. The 5 speed lathe I got is on the better side of that equation, but it is still medium quality at best.
Having said that, it has enough power to turn pens. It is an MT-2 machine and it has minimal runout. The factory tail stock center is junk, both in bearing quality and the taper angle. Replace it immediately. I don't mind switching belt settings and I don't know if I would trust the Harbor Freight electronics for variable speed. My next lathe will have variable speed. The tool rest is OK, it helps to clean it up with a file, as the casting/machining wasn't great. I've been using it for about a year and nothing has broken or gone bad.
Personally, I believe that the sharpness of your tools is the most important factor in turning once you develop some basic techniques. My wife exclusively uses carbide insert tools and I use standard HSS tools 85% of the time. She's better with hers and I'm better with mine, but I sharpen often and she doesn't at all.
Also, in my opinion, the second most important thing with pen turning is the squareness of your cuts on the prepared blanks. There are a bunch of ways to get them square, but make sure you understand how to do it and that you do it well. Unsquare cuts leads to a temporarily warped mandrel when you tighten it and the result is off center turning, which lowers the quality of your pen significantly. Many people swear by turning between centers, which removes the issue, but another way to remove the issue is to square your blanks well to start with. Read the forum and decide how to proceed based on what you read, but I assure you that using a mandrel with poorly squared blanks will yield poor and frustrating results.
I use my drill press as a pen press and it has worked fine. If I had the choice between buying a pen press or putting the dollars towards a Craigslist drill press, there would be no question: drill press.
As a new turner, I'm sharing the experiences I've had over the past year and a half. This is an accessible hobby if you don't go off the deep end at first and buy all kinds of things that aren't needed.
Lastly, check the chapter forum to see if there is an active IAP group in your area. They could answer your questions in a hands-on way that is hard to beat.
Good luck.
Sandy.