Cheapest for the first 1-2 upgrades I think would be spamming silas stash quest on darkmoon faire with a bunch of level 15+ toons. You get 100 dmf tickets off that which I think can pay for 1 upgrade each time you do that quest. Quest only doable once per toon though.
Krom at the library in hall of explorers in iron forge sells upgrades for heirloom armor and weapons. Also I think a lady above the front gate of orgrimmar whose name I forget.
The heirloom upgrades will cost thousands and thousands of gold, so you should focus only on maybe one plate, one mail, one leather, one cloth and not try to get everything. Just whichever one set your current levelling toon needs.
You’ll see upgrade level 0/4 on heirlooms to start. The first upgrade makes them go to level 90, second to level 100, third to level 110, fourth to level 120.
Also you should be aware that after shadowlands drops and level squish happens, the heirlooms will start out at common on 0/4, then upgrades will make them go to uncommon, rare, and epic quality (?) feel like I’m missing one there… In Shadowlands you will also be able to purchase sockets and enchants to put on heirlooms in your collection. The exp buff will go out of heirlooms and just be universal for all players after SL drops too.
So, just upgrade maybe the one set you’re using to level right now. And maybe only do the upgrades as you hit level 60, 90, 100, 110 instead of doing them ahead of time. A wow token might go far upgrading one set of heirlooms.
Also I have an Alliance main on Dalaran server who’s in a guild that has all the heirloom pants unlocked. If you just roll a whatever over there you can whisper me for an invite so you can get the pants. My battle tag is SilentNight#1412 and I’m usually on between 6pm and midnight EST on weekdays.
Edit: Forgot to mention if you roll a new toon on a new server and they’re alone and poor. You can have a rich main from another server buy some battle pets off the AH for thousands of gold, then add to collection. Then have your new alt on new server open the collection, cage the pet, and sell on the AH on that server. You might plan on selling the pet for half or 2/3 the lowest price you see if you want to sell it fast. This guy I’m posting with has been trying for a week to move a Syd the Squid, but I’ve only done a 20% markdown against competitor so far.