I disagree award booking service is necessarily the way to go.

If you want to do this once, then yes, use an award booking service. But I think a better time investment is to learn the ropes and ins and outs of stopovers/open jaws/different rules. This is not the last trip you're taking and the time investment will be well worth it in the end. You'll know where to collect miles for next time and being able to construct bookings yourself will open up far more options.

Many of us will tell you what you need to know for free and help you walk through the award booking process. Don't feel overwhelmed. Each airline forum has threads on routings and stopover rules so feel free to read through those and post there. Since you have United miles and Ultimate Rewards points, I suggest you start with the Stopover/Open jaw rules thread in that forum and I can almost certainly personally guarantee that many of us are happy to answer any routing questions you have once you've read a little bit.