When choosing a filter for a axie, one must keep in mind that axies don't like water movement. And in saying this, I'm a huge fan of the basic air powered sponge filter. They take a while to get established, but couple them with regular water changes and monitoring of the water qualiy, and they work a treat. The also don't add any heat to the water like a power filter would.
If your tank and budget is large, canister filters are a fantastic, they are very low maintenance, and provide powerful biological filtration as the sponge filter would. I have a aqua one canister filter running on a 6ft tank and it's been going for years, and it gets cleaned only when the flow starts to diminish, so every couple of months. As for choosing a filter size, you want one that turns over the entire volume of your tank about 3 times per hour. So if your tank is 100L, you want a filter that is rated for 300L per hour. The box will state what size tank the filter is recommended for.
My axie setup is a unusual one, I home a single female axie in a 95L / about 25G "aquapro" tank that has a built in trickle filter in the hood. The top of the filter box is open, and I grow peace lilies right in the filter box itself. The plants suck up all the nasties my axie excretes and I always have fantastic water qualiy. If you like, I'll upload some pics of my setup for you.
As for hides, you want at least 2 per axie, so they can completely hide away if they wish. My axie does not use the hides I have provided much, but I leave them anyway just in case.
And if you can't resist buying a axie until your tank cycled, you can keep one in a uncycled tank, but you will need to test the water daily and monitor the chemistry and change about 30% daily. It's hard work, but although not usually recommended, it can be done.