It is worth checking via a google search, but I reckon your palm is a Cordyline. Detach the new suckers in march or april and pot them up with John Innes No 2 compost. grow them on in pots for a year, then you can plant them where ever you want. Or you could just leave them all where they are.

You can only grow those new shoots on if you can get a good piece of root with each one - usually they arise out of the base of the trunk, so it's often not possible to get root with them. Chop down the remaining trunk to the new growth and let them grow on into another, multiple branched Cordyline.