The fastest way to gain electrum is to win against FGs and sell the upped cards you win (at roughly 1100 electrum per card). You can try doing this as soon as you finish all the quests, but your chances of winning are a bit on the low side because you'll have to run an unupped deck.
Some good choices for relatively unupped FG decks include:
Evil Twin (
http://elementscommunity.org/forum/index.php/topic,22541.0.html) (~30% win rate, completely unupped, will probably decrease significantly once the AI understands how to deal with Cloak)
Flay 'em (
http://elementscommunity.org/forum/index.php/topic,20228.0.html) (~30% win rate, 4 upgrades; works completely unupped, a solid choice overall)
RoL/Hope with 6 upgrades (
http://elementscommunity.org/forum/index.php/topic,14533.0.html) (~25% win rate; fast games, fast autoquits)
Liquid Antimatter (
http://elementscommunity.org/forum/index.php/topic,17730.0.html) (~15% win rate, completely unupped; fast games, fast autoquits)
Rares are available most reliably from farms in T50, though you still have small chances to win them in spins from AI3, AI5, and FGs (depending on the rare in question).