They go first, but they pay a lot of points for models with low toughness and poor armour, if they leave anything alive then they die. And with our very poor initiative we should expect and even rely on the fact that we will not strike first against any foe, then if by chance we come across one where we can go first it is a bonus.
Saurus warriors will have absolutely no trouble with spearmen even with their strike first, in fact I would go as far as to say 15 saurus could be 20 spearmen. Assuming neither are charging for maximum attacks on both sides, the HE get 15 attacks; 10 hit, ~4 wound, 2 die. Saurus retaliate with 16 attacks; 8 hit, ~5 wound, 3-4 dead elves. They have a rank up and outnumber, 15 saurus win. Doesn't look so scary now does it?
Sure swordmasters and white lions have higher S and WS so will deal more damage, but they also cost a lot more points and die very very easily to shooting and magic. Also, impact hits go before ASF so the stegadon will have a field day.
But I digress. If you get the charge (so they don't get charge bonuses) you should survive, I probably wouldn't attack dragon princes with a lone chief though. They have 2 attacks each.