The speed of the axies determines who goes first. Each axie has a base stat based on base body type. The base stats are Health, Speed, Skill, Morale. Their sum is the same but the proportions are different. Each aqua and bird card adds 3 speed, each beast and reptile card adds 1 speed, each bug and plant card is 0. Values start at 31 (plant and bug) then 35 (dawn and beast). Pure cards make these axies a specific speed, 61 for bird, 57 for aqua, 41 for beast and reptile, 31 for plant and bug. You can use different cards to manipulate this but you might lose the card bonus value (10% attack or 10% defense) so you need either class advantage cards or chain card that are at least 5-10% better or higher utility. These are important breakpoints, people try to get higher speed axies for pvp, so reptiles are often 42 or more speed, aquas on 57 are having a bird part, aquas with beast or other cards exist, but no lower than 53 so they can still speed above birds.
Speed determines turn order. Faster Axies attack first. If two Axies have the same speed, this is how attack order is determined: High speed > Low HP > High Skill > High Morale > Low Fighter ID
You can see the order of axies on top of your screen, it applies to the next turn once all cards have been used. This means that 2 aquas on same speed and undamaged, the one with a bird card goes faster, so 5/6 purity aquas are better, usually eyes and ears or horn cards are replaced. (bird has 3 speed like aqua but no health bonus, only 1 morale).
If your axie is damaged, he will attack first on same speed, this is a good thing to consider, and let them attack you, so you withhold some damage but increase shields, and finish them off next turn.
Without any bonus, birds are the fastest, and they can have 3 high damage cards and a 4th 0 cost to kill a front tank turn 1. Aquas however are faster with the speed bonuses.
If the axies werent attacked yet, and they are the same cardset and purity, then often the HP, the skill and morale matches so the ID number decides, axies born later have higher number so they let the elders attack first. Eyes and ears might be hard to see ingame but they got the same effect. While low ID birds can even be more expensive, and low ID aquas aren't that searched for but still got a bit of advantage (a single or more bird part still faster), tanks need higher ID to go last and steal back energy or generate safely.
Speed is a good thing when it comes to attack, but it takes away from the other stats. Also the way the energy generating, stealing and destroying works, you can't steal or destroy energy if the opponent spends it all, if they go first and gain energy you will be able to steal or destroy. That's why for these skills, going last is better. And since the plants got a few energy cards, and they go last, the slower plant is the better. For everything else that does damage, going first might let you kill the opponent and waste his cards and energy. There is also a tradeoff, arco or incisor aquas are getting in meta, some cards got more damage against plants like dual blade or arco, but they slow down the aquas base speed (beast part 1 speed 3 morale), they won't be able to match other aquas without speedup, but they already need a speedup to match birds. So aquas above 52 speed (52x1.2=62.4) are still viable just need a speedup, but on the other hand they get utility or damage.
There are 6 cards that interact with speed, these cards apply for the next turn not the current, so you need to survive to use them. There are 3 cards that speed your axie up: goldfish, koi and arco. AAP players are way too familiar with the power of these cards, since most aquas are 420 health and under 100 armor, but deal 4x120 to 4x140 damage, the one who goes first wins. In case you cnat kill each other in a single turn, the one who uses more speedups in attack (koi) or more speedups in defense (goldfish) wins. That's why this cards are far the most popular cards while breeding and buying/selling aquas. Arco fishes are getting popular as they can use a beast damage advantage against plants instead of their -15% aqua cards. They lose 10% from same card bonus but still 5% value gain and 50 shield which is higher than any aqua card that has damage on it (only sponge, anemone and hermit would be more that is usable and popular). You can speed up 5 times, each is 20% step (round down? I think).
Speed downs are the exact reverse, making your opponent slower, 1 speed down=1 speed up. It won't take 20% of their final speed, it applies to base speed. maximum 5 speed down, for a total of 100%. They will go last even if they are birds against a plant. So goldfish can resist a terminator this way.
Speedup is also a buff, so Iguana activates on it : "Gain 1 energy when attacking a buffed target." There is only attack up, speed up and morale up that is a buff coming from cards.
Laggin: important part of the terminator combo, it's a 0 cost that helps activating the 2 card combo, and speeds down opponents, especially useful since reptiles are slower than aquas, especially using bug cards. Croc: a really high value ranged card, saw some people using with cuckoo. Bonus that speeds down opponents. Incisor: also high value card that speeds down opponents.