I have purchased a Garmin for my son who doesn't have a mobile device yet. What is the best way to set up his device on my phone so that we each have separate data?

AFAIK, there is no way to do it on a mobile device. The only way to have two devices synced to two different accounts is to use Garmin Express.

If you have an Android tablet with multi user support you could probably use that and let him have his own user.

Just to add to what tmk2 and e7andy have mentioned. If your son is under the age of 13, there is no "legitimate" way to setup their own Garmin account. Since Garmin is US-based and due to the U.S. Federal law Children's Online Privacy Protection Act, 

That law only applies for US users. Other nations have their own age restrictions.

Thanks for the clarification. I'm not a lawyer, but reading the wikipedia article...

Applies to the online collection of personal information by persons or entities under U.S. jurisdiction about children under 13 years of age including children outside the U.S., if the company is U.S.-based. 

Assume if the law is more strict in the UK (16 years of age), then that law would apply. At any rate, there should be some sort of age restriction.

Out of curiosity I went through the list of countries and they all say 16 except the United States which is 13 so I don't think for the UK that UK law has anything to do with it. (I don't know what the age limit is for online services in the UK is but if it is 16 then I do know there is a lot of law breaking going on on a daily basis.)

I just spoke to my wife, and I'll add my son's watch to her phone ("fortunately" she doesn't use a garmin watch yet).