Should I update vCenter Server and Update Manager first, then the ESXi hosts?
First thing you have upgrade the vCenter and VUM and then the ESXi host.
In vSphere 5.1 you have additional components which needs to be installed SSO and Inventory Service before you can upgrade the vCenter Server.
Suggest reading the Upgrade guide (http://pubs.vmware.com/vsphere-51/topic/com.vmware.ICbase/PDF/vsphere-esxi-vcenter-server-51-upgrade-guide.pdf)
Also make sure the hardware is listed in VMware HCL and make a practice to check VMware Product Interoperatibility matrix