You only have one year to obtain your UK license. The earlier you jump through all the hoops, the better.
Yes, you'll want to get started on obtaining your full UK licence as soon as you can. You are only legally allowed to drive on your US licence for 12 months from the day you arrived in the UK on your spousal visa, and so if you want to keep on driving after the first year, you must have passed all the UK driving tests and received your UK licence by the time the 12 months are up.
First thing to do is apply for a provisional UK licence (which can also be used for ID instead of your passport). This can take a few weeks and I believe you will need to send off your BRP in order to get it. See here:
https://www.gov.uk/apply-first-provisional-driving-licenceIt can take several months to pass the UK tests and get your licence (and it may take more than 1 attempt to pass the practical tests as they only have something like a 45% pass rate), plus I would recommend taking some driving lessons to learn how to pass the test, as there are some very specific things you have to do in order to pass (even down to how you hold your hands on the steering wheel, how you check your mirrors and the exact distance you are from the curb when doing certain manoeuvres!).
If you are unable to get your UK licence by the end of the 12 months, you will be treated like a learner UK driver, which means that until you get your UK licence, you must display red L plates on the car at all times and you must be accompanied in the car at all times by someone over 21 who has held a full UK licence for at least 3 years.