Should I buy a GPS?
Maybe (or rather probably yes), but you don't need one to estimate elevation gain or loss, and they may not be very accurate for this purpose. There are good reasons to buy a GPS; I use one mostly for fun (geolocating photos with a track), but a track can also be a life safer to retrace your steps if you get lost, and a dedicated GPS may produce a longer and more accurate track considering battery life. If you do estimate elevation gain or loss with a GPS, make sure to download the track into a computer and remove measurement errors first, as a couple of bad points can easily inflate the integrated climb estimate significantly.
Or are there cheaper alternatives to calculate my elevation that is reliable?
Canadian topographic maps are available free of charge through the National Topographic System. They cover the entire country (!) at a scale of 1:50k, but may not include all hiking trails in remote areas. They may also be decades old and in black and white. In popular areas such as Jasper or Banff National Park, commercial alternatives exist such as by National Geographic or (in slightly less popular areas) Go Trekkers (they reportedly also sell enhanced versions of official topographic maps, adding shaded relief and colour where applicable, I have no experience with those). Their scale may be less good, but they may be more complete for currently maintained hiking trails (they may or may not include decommissioned or unofficial trails).
It is easy to tell elevation gain or loss from a topographic map if you are competent in reading them. Competence in reading a topographic map is certainly a prerequisite to become a mountain guide.