The greenest home is usually one that already exists. Most existing homes have room to become more efficient, and the question is normally how to pay for the work without wrecking the monthly budget.
Air sealing and insulation are generally the first thing to do, because they are comparatively cheap and they reduce the load on whatever heating and cooling system you install afterwards. Doing it in the other order means paying for a larger system than the house needs. A heat pump replacing electric resistance heat usually cuts heating energy use substantially; replacing an efficient gas furnace is a much closer call and depends heavily on local electricity and gas prices. Rooftop solar economics turn on your utility rates, your roof orientation, and what your utility pays for exported power, so it is worth pricing rather than assuming.
Savings figures quoted for any of these are averages across very different houses. The only numbers that mean anything for your decision are your own utility bills before and after, and a contractor quote you can compare against them.
Three financing structures come up repeatedly. An FHA 203(k) rehabilitation loan bundles improvement costs into the purchase mortgage, which suits a house you are buying and improving at once. A HELOC lets you draw as projects proceed, at a rate that is usually variable. A cash-out refinance suits a single large retrofit, but it re-prices your whole mortgage to do it, so it only makes sense when the new rate stands up on its own.
If you are weighing the cash-out option against keeping your existing loan, run both through /mortgage-calculator/refinance/ before committing to the work.