You have a fair chore here... it will take a little work and some downtime (although not significant). Here are some general steps assuming you are NOT using the embedded switch in the CXi, if you are it is a bit more complicated.
In general, you do not change the VLAN information in the Mitel, you will change it at the switch level.
First create your VLANs on both switches then set the switches so the default VLAN on all ports is 350, and 300 is allowed as tagged traffic. find the port the Mitel is on and change it so that 300 is default VLAN (PVID) on that port.
Log into the Mitel and change the IP address and DHCP information (remember NOT to define VLAN information here)
Those switches support Auto Voice VLAN, so I would just set them up so that 300 is the Voice VLAN and add 08-00-0F to the MAC table for voice devices (if it isn't there), then you don't have to mess with DHCP on the data VLAN in the Cisco (I am unable to help with that if it is essential to do it that way).
If you plug into a port on the switch with a phone, it should automatically be placed into VLAN 300 by the switch and request DHCP in VLAN 300, receive it from the Mitel and come up and work.
This will not allow you to talk between VLAN 350 and 300 though, they will completely distinct... if you want them to talk to each other, that is for someone with a little more router knowledge than me.
TBH, this is a fairly complex project, not extremely difficult but there are lots of little pieces that have to fit together just right for it to work. I would suggest contacting your IT support and Mitel vendor and get them together to come up with and implement a plan. If you really want to do this on your own, I hope you have a fairly extensive knowledge of Cisco and network in general, especially with VLANs.
