Hi Helen - welcome to the fold.
The two mandrels mentioned above, the Planet and the Axminster are the only two I would recommend, and I've tried several in my time. The planet requires a 60 degree live centre in the tailstock to support it, one of which is usually always supplied with a lathe when you buy one. The the Axminster comes with a "mandrel saver" which does away with the need for the live centre.
The Axminster is by far the more popular, which may be down to the size of Axminster's marketing machine. It's a good quality piece of kit but the downside of it is that it's shaft is possibly the thinnest on the market, which means that bushes from most kit manufacturers will rattle around on it to a greater or lesser degree. I personally use the Planet which is also good quality (it's actually a Rotur, which is the same thing) because it has a better sized shaft.
For myself I would avoid most other mandrels especially the ones available from any number of re-sellers of cheap chinese pen kits - the ones where you can get a slimline for a couple of quid or less. It's not a problem with the re-sellers but the mandrel they all sell is supplied by a firm called Rizheng and I have never, ever come across one that is straight, which is a pretty fundamental requirement in a pen mandrel.