Insulate, new double panel windows, and heat pump if you can afford it.
Insulating should be the first option to be consider, but it can be a costly renovation. Spray foam insulation is a good renovation option because the wall doesn't need to be remove for bat & plastic sheeting.
And, keep in mind that insulation thermal resistant (R-value) return works on an exponential scale therefore at some point the R-value return doesn't increase significantly enough to warrant the increase in insulation volume/cost.
Heat pump is not worth the renovation cost for a house that have existing electric heating or hydronic heating (geothermal heat pump is an option with existing hydronic if the soil/large body of water is favorable).
Air source heat pump is the way to go, if the house have existing oil/gas heating ductwork (at -1 C or -2 C backup heat source kick in). Oil/gas ductwork is slightly undersize for heat pump but it only need minor modification on the return air (larger return air opening) to accommodate the heat pump cfm requirement. However, the supply line will be undersize therefore the friction air speed will be higher (translate to slightly higher blower noise transfer, make sure to get variable speed blower to accommodate/prevent back pressure against the blower).
Air source heat pump best case scenario for energy saving can be up to 75% less (manufacture & pro heat pump organization/business) than traditional oil/gas/electric energy consumption. In my experience air-to-air heat pumps saving is around 55%-60% (at perfect ductwork calculation, heat pump sizing, perfect refrigerant level, and mild coastal climate) when compare to oil/gas/electric.
Geothermal heat pumps manufacture claim up to 85% energy saving compare to oil/gas/electric energy consumption, but in my experience at best it is around 75% in energy saving compare to oil/gas/electric. Ambience air temperature don't affect geothermal because the heat is ground source. (High initial setup cost)
Where can I send you the bill for the consultation fees
Some websites seem to indicate that natural gas would be about the same cost as operating a heat pump. I have also heard some people say a heat pump is not too great paired with a fossil fuel furnace because the coil gets too warm for the heat pump to be effective.
It greatly depends on your local natural gas price vs. electric price, but for most milder climate (coastal area) air source heat pump operational cost should be at least 1/2 that of NG.
It is fine to have fossil fuel running as backup with heat pump, and in many case better if you have backup electric generator, because fossil fuel heating require very little electricity to operate in a electrical blackout.
Adding 20" of insulation yourself in the attic should be less than $1000 if you do it yourself which is very easy. You need to make sure though that you have styrofoam baffles installed against the roof though to allow airflow.
Double the thickness of insulation is not going to double your R-value. If the R-value is already at 45-50 (14" fiberglass bats, or 12" blown in cellulose) you aren't going to get much energy saving when doubling the thickness of the insulation/cost.
We're wondering whether we should get a new gas unit, and whether it might be better to go 2-stage with a heat pump. We got roughly the same seller subsidy for that same purpose
Go with heat pump/fossil fuel if you have the finance.
Make sure they don't go and buy a builder's grade heat pump - it won't save as much money as something with a good energy star rating.
In some case you are correct, but in others you are not.
It all come down to support & maintenance. There are only a handful of heat pump manufactures but there are many make an models (many are manufacture under different names), however the refrigeration/mechanic companies specializes in certain make/models for simplicity sake (ask for high SEER value). Therefore, it could be cheaper on install & maintenance fees if you stick with the builder/installer suggestive models.
I have seen many cheap heat pumps (thousands dollar less than builder/installer) that sells direct to the uneducated consumer. Cheap heat pumps tend to have slightly cheaper compressor (nosier and doesn't last as long) and/or poorer workmanship. Some weren't intend to be repair, because there aren't space to work on it or the parts are facing the wrong way/difficult to get at, therefore the high repair cost can be very intimidating (there are mechanic that refuse to work on lesser heat pump because they do not want to warranty it).
There are good deal on heat pumps to be had on the internet if you are an intelligent consumer that do your home work. But, keep in mind the maintenance and installation factor when you do your shopping.