Experienced cyclist here. What kind of bike are we talking about? Is this a road bike/hybrid/MTB/etc.?
Where did you get the bike? If it was from a shop, did you get a fitting? I assume the shop made sure it was the right size but a fitting is still important – they will do a much better job than you can by looking at on line calculators. Investing in a basic fit (about $65 or so) from a pro fitter is worthwhile (if it’s a road bike).
Going back to your OP, I’m confused about your MD’s advice. If you lost inches but gained weight, that tells me that the weight gain is due to more muscle. That’s good. I mean you don’t want to lose the muscle. Sometimes, a person’s weight isn’t a good measure of health or fitness. But whatev.
Using the trainer isn’t going to help you with your balance issues so I’m not sure what you mean when you say you “want to get better before hitting the pavement.”
In terms of the HRM, research how to use it for cycling purposes. The zones are only good to use for recovery rides and base building. As a beginner, you are probably at base building (look that up). Forget the varying your speed thing until you are more experienced and have a good base built up. After that you can start learning about different intervals, different workouts, etc. There are plenty to choose from depending on your level and goals. You can also then begin to learn how to use the HRM (you’ll need to determine your Lactate Threshold Heart Rate and use that rather than the 5 zones), but that’s for the future and only if you want to really train). Oh, and speaking of zones, make sure to not use that goofy 220-age formula, or any others. Determine your own max HR. The formula’s have high standard deviations so they aren’t useful and can be very imprecise.
For now, I would suggest that you JRA (Just Ride Around). No training plans or anything like that. Just get used to the bike and find out why people enjoy the sport – a large part of it is nature so doing the trainer won’t really get you there. We do rollers and trainers as a necessary evil to keep our fitness base for when the season hits.
Just ride, nice and steady and let your body acclimate and find its space on the bike. Get more comfortable with balance, control, stability, etc. Learn about efficient pedaling technique and cadence, learn about the importance of hydration and nutrition. Once you get a season or so on the road, a bunch of hrs, and a few thousand miles into the legs, you can start upping the intensity and researching training and learning a thing or two about suffering. And don’t forget about group riding (when you are ready). It’s a whole different animal than a solo ride.
Best of luck.