As with anything, there is a trade-off between price and quality, but more expensive isn't always more better. I had some tools to start with, but I will give you my thoughts:
Lathe: mine was gifted to me used, but is valued at about $400 new. Variable speed is nice, but not necessary, and adds some cost. Actually, if you drill on the lathe you will avoid some problems if you don't have variable speed (but you will need to change belt positions to change speed). Mine is not a variable speed, and it doesn't bother me. You will need chisels too, lots of options, I got HSS tools with the lathe but spent $150 on a sharpening system. Carbide might be a better place to start.
Drilling: a good drill press is expensive, and cheap ones don't work very good. You will get better results using the lathe to drill, but you will need some attachments for your lathe. A Jacobs chuck (about $20) to hold the bit, and a chuck to hold the blank. There are lots of options for chucks to hold the blank, I got the Nova G3 ($100), I also got the JSPIN jaws ($40) to better hold the blank.
Trimming: most videos on YouTube they use a barrel trimmer with a drill press to trim the blank flush with the tube. Since I opted to not get a drill press, I went with a disk sander for this task ($100). There are other options out there, this was just my choice.
Mandrel: I recommend going straight to Turning between centers (TBC), you can find a lot of info on IAP on this topic. All you need is a dead and live center. The TBC bushings are a little more expensive, but since you haven't already purchased standard bushings the added cost will be minimal. I started with an inexpensive mandrel and wasn't happy with the results.
Other stuff: a hand saw will get you by with cutting. I personally think a table saw is the most versatile saw for woodworking, I have a contractor saw that has been very useful ($300). If you go with TBC, and you want to do high-end pens, you will also need a decent digital calipers.
All said and done, I think you can make a pen with an initial cost of about $600. But a lot will depend on your choices. New vs used tools, high end versus budget, power vs hand tool, etc.
I hope this helps, best of luck!