The best way to play the civ is to abuse every opportunity that it creates for you. That's not what you've been doing from the sounds of things. You're probably playing too passively and you're probably ignoring religion. Songhai is an aggressive civ and if you're not actively conquering lands then you're not putting him to use. First of all, you might want to rush Philosophy and abuse Legalism to give your civ some free MPMs. Your UB produces culture, so if you have Monuments in your first couple of cities when you take it and you have Philosophy researched then it'll give you those. The faith is what you're looking for at this stage in the game more-so than anything, so getting those "Temples" immediately is ideal. I know that a free Amphitheater sounds better since it saves you ~6 gpt or whatever, but you really want that faith rolling asap. For your religion you're going to want to rush Holy Warriors as your follower belief. Remember, your UU has a reduced production cost which also translates to a reduced faith purchasing cost. Your immediate "Temples" paired with that savings will allow you buy a bunch of Calvalry which have no city penalties. From there you should go on a rampage and take out any nearby satellite cities. If you need siege weapons for Capitals, that's fine, but your Cavs should be fine for the expos. I would have your cities building Archers/CBs early on and worry about buying your Cavs with faith. You'll still have to make/upgrade some Horsemen obv, but faith will do some of the work at the very least.
The UA is "fine" and can help you do things like quick-buy workers and settlers sooner than other civs can. It's also a great way to maintain a large standing army when you're capturing cities. You don't always have 7 turns to wait around for more Archers to come, buy being able to buy a new "something" every time you take a base goes a long way to keeping the war effort moving along.
Still, I highly recommend the Legalism "trick" (if you can even call it that) and Holy Warriors to buy a lot of Cavalry. The hammer reduction may not seem like much on paper, but it's actually quite relevant when you pair all of this together.
i just tried an immortal/continents/standard speed/standard size map with songhai and didnt get very far, if you could add comments i would appreciate it:
i went the standard tradition 4 cities opener (buying 2 settlers) but held off on legalism untill i had shrines/monuments and philosophy. i also had libraries quickly and made NC in my capital. it seemd like a great start
i was located in between babylon and siam (who was going OCC by the looks). babylon had CBs before me even tho i went for them fast, but i managed to kill his invasion. However, he had walls of babylon in his cities so i figured my 5 cbs and 2 spears are not match, we made peace.
from philosophy i rushed to chivalry ASAP, whilst building several horses to upgrade. however, before the time i got there, siam already had his elephants and attacked my bottom city, which i held off. however, babylon later attacked my top city just as i got to chivalry, and his trebuche and mixture of forces was too much for my split army.
once i popped legalism i was generating 12 faith per turn, but only my second GP was born before chivalry and i didnt have enough faith built up to buy the cavalry. is this enough faith?
basically, i had a strong start but got tag teamed by babylon and siam. Oh, also just before i reached chivalry babylon built the great wall :/
any suggestions or things that i may have done wrong or overlooked? is 12 faith per turn really enough to justify going holy warriors?