To convert a video tape to DVD you need 1) a working VCR and 2) a device that will convert the VCR signal to digital format. #2 can be a DVR (some of us have Magnavoxes or other brands with hard disks), a video capture card inside a regular computer, or an external device that plugs into a computer. I have used both a DVR and a card inside my PC.
The nice thing about all of these methods is that they also allow you to record and make DVDs from new broadcasts by connecting your choice of device #2 to your cable box (or cable, or antenna), though copy-protected broadcasts may require workarounds. Programming a DVR is a lot like programming a VCR. Timed recordings can also be set up on a computer with the right software. The computer-based techniques make editing (like removing commercials) easier and more precise than using a DVR, which is helpful on both new broadcasts and videotape conversions.
It's not hard to convert tapes to DVD, but the quality you get from VCR tapes is not good unless you recorded just 2 hours per standard tape (SP speed) and your tapes are still in good shape. I'd recommend attempting conversions only for things not available commercially on DVD and not being rebroadcast, such as old sporting events or classic TV shows not showing up in syndication.
I'm not admitting how many movies I have on DVD. Let's just say I'm glad I have a spare bedroom. Recording movies can be addictive. I've finally reached the point that there's not a lot of new stuff for me on TCM--which I'm not sorry about, given how many unwatched movies I already have on DVD. However, I still use an Excel spreadsheet to keep track of what's coming up so I can figure out what will be recorded on each device. I list date, time, channel, program name, IMDb rating (movies only), duration (movies only, in case it helps eliminate overlaps). I have a separate column for each recording device (no, I'm not going to tell you how many of those I have, either) so I can plot equipment usage. This allows me to monitor availability of reruns (popular on PBS) so I can deal with recording conflicts.
By necessity I'm doing better about watching things rather than automatically burning practically every movie I'm interested in to a DVD, but I'm a big believer in getting things off the DVR or computer as quickly as possible so I don't stand to lose too much if there's a hard drive failure. Anything really important to me gets put on a DVD quite quickly, which I then proof (fast forward through) to verify completeness. Just yesterday I caught an EBS alert marring La Ceremonie, and I was lucky enough to find that movie still available via TCM On Demand.
I maintain a list of DVDs, alphabetized by title, that shows source (channel), name, whether I've watched the program or not, and any important reservations about quality of the video (such as the edited length if the movie has been cut). I now also include movies and programs I've watched rather than converted to DVD (marked as such), because I found I was recording things I had already seen. I try to remember to check this list before setting up any recording.
I have a separate document on which I track episodes of TV shows I'm keeping, so I can be certain they are complete.
I store the physical DVDs in thin jewel cases. Although I started with everything alphabetized together, it made for an awkward number of boxes, so I decided to split out a few categories: travel shows, cooking shows, documentaries, musical programming, and TV series. That has helped a little--those categories are much more accessible, but the boxes of movies and miscellany are still a problem. Because of the effort required to shift boxes around, I let a stack of movie DVDs build up for quite a while and file the whole bunch at one time.