The basics - wetsuit and trainers.
Wetsuits - most swimruns require you to wear a wetsuit but there are a few in warmer climes where it is optional. You don't need to shell out on a swimrun specific wetsuit for your first race. Many just take an old triathlon wetsuit and cut at the knees and elbows (depending).
Trainers - there's a variety out there, key considerations are weight, grip and drainage. For the more mountainous races you may want to consider more support and cushioning. WESWIMRUN can recommend Inov-8.
There are few rules about what extra gear you can race with. For your first swimrun it's a very good idea to keep it simple. The more kit you have the more you have to manage. In training allow yourself time to experiment with different gear, make mistakes, learn and evolve.
Remember each swimrun is a different distance, different swim:run ratio, and a different learning experience. This is one of the reasons it is so engaging.
Increase the bouyancy of your lower half by using a pull bouy. You can also play around with ideas to increase bouyancy of lower legs, neoprene calve guards are used by some.
This is effective in the swim for keeping together and speeding up a slower swimmer. It can help on the runs too. A simple length of bungee with 2 clips attached to both partners works. Practise with your partner!
Lots on the market, different sizes, shapes, attachments - try some. Consider the length of the longest swim, transition management, and how you will store them during the runs. If you are going to use them, train with them first! Some paddle advice here.
Key things to consider when making your decisions: ratio of running to swimming; longest swim length; longest run length; water temperature; air temperature; wind. Kit management can add stress to your race if you are not used to it, so for your very first swimrun it's not a bad idea to strip back to minimal kit, especially if you haven't trained with your extra kit.
Remember every swimrun is different just like every human is different. Part of the fun is deciding what's right for you (and your partner) for a given race, on a given day.