How to start paying US taxes as a dual citizen

I am a UK/US dual citizen; born and living in the UK. I have travelled to the US several times, but never lived there.

My family’s attitude towards US taxes has always been "stay off the IRS’ radar". This makes me feel slightly uncomfortable, especially since (afaik) the UK/US tax treaty means no one should end up paying more tax than they would have as a citizen of just one of the countries anyway. I understand that there would be additional complexities (work) involved in filing twice, but (to me anyway) it seems like it’s worth the effort to avoid the possibility of being caught for years of back taxes sometime in the future.

I’m currently a little over a year out of university, having been full-time employed for close to a year.

What (if anything) should I be doing to correct my historical lack of US tax filings?