Welcome, and be prepared to spend a lot of cash on this hobby. It is fun, and pretty easy to get started and a novice can achieve really good results. If you do not own a drill press, I would not invest in one for pen blank drilling. Most bench drill presses have insufficient quill travel to drill long blanks, and so a floor model is better suited. And keeping the hole centered in the blanks from end to end can be very tricky. rather than invest in a decent drill press and vise to hold the blanks, doing them on the lathe is easier. Before I got the pen blank jaws, I turned the blanks round, and then conventionally chucked them in a lathe chuck to drill them. The jaws eliminate the need to round them first, but either way, you woudl need a chuck to hold the blank for drilling. I own two Nova chucks, since I also turn peppermills and other larger objects, and need a variety of ways to hang onto a blank for turning.
BTW - I do all this on a Rikon mini lathe, a belt changer (6 speeds). But I did add a bed extension so I could turn and drill longer objects. It is a good lathe, although to get variable speed requires a user-installed upgrade kit.
I have a drill press I use for a myriad of tasks, but no longer do I use it for pen blank drilling. I just bought a set of pen blank drilling jaws for one of my Nova chucks. These do an excellent job.
I also bought a Woodchuck Pen Pro carbide tool last year for use on pens, and really like it. But I still find myself using basic tools like roughing gouges, spindle gouges, scrapers, etc. I'd recommend getting a budget set of tools, and learn to turn with them, as well as learn to sharpen with them. I have a full size set of tools, and not the smaller pen tools often sold. I find the pen tools just too small to use comfortably.
I've been turning for about 3 years. I wish I could say I've mastered the skew. It is a tool not for the faint of heart when new to turning. It is one on my goals this year to became "one with the skew".
Good luck in all your purchase decisions!