There are a couple ways to get XP for your mage units. The easiest is simply to build Adepts early and often. Use them against barbs, but try not to let them die... it's usually better to suicide a Scout and make it a sure thing than hope your Adept can win at 60%. Even without XP farming, if you built enough Adepts and didn't let all of them die off right away, by the time Archmages are available you should have some with enough levels.
Otherwise, there are some Wonders and Civics that grant free XP. Combine the Combat civic with Apprenticeship, and you have four free XP. Both of those are easy to research. Getting Theocracy and the XP Wonders are generally a little harder to get to, and require more time.
Neither of those options is the "quick fix" you're looking for, but really, that's the point. Getting an Archmage isn't supposed to be easy. That's the whole point of the Amurites - they are able to get their magic units from Adept to Archmage in a rermarkably short time. The Potent promotion and the Cave of Ancestors, combined with the aforementioned Civics, generally allow for a new Adept to start with enough XP to become a Mage, and get to Archmage status in a couple dozen turns. (That is, they allow for it when handled correctly - you'll want a lot of mana).
Of course, and this is a bit off-topic, it can be argued that Archmages aren't worth the effort. The spells they offer are pretty good, but you'll find that some of the best ones are actually at Mage-level or even Adept-level. The terraforming spells are for Adepts-only; so is Courage and Raise Skeleton. Regeneration is for Mages, so are Fireball and Blinding Light. Most of the Archmage spells are summons. Don't get me wrong, summons are very nice to have - but it's much easier to get Iron-using Champions than Air Elementals, and your Champions can keep the promotions they gain. This is an oversimplification, but the point is that getting high level magic users can be more risky than the reward is worth.