There is a good deal that could potentially be going on at boot. But, it seems that you are interested in what happens when to get a DHCP address, so I will speak to that specifically.
First, when a machine comes up to the point that the DHCP client is ready to get an IP address, a DHCPDISCOVER message will be broadcast to the local subnet. If there are any DHCP servers on the network they will respond with a DHCPOFFER to the client. This will contain an offered IP address and a lease duration at the bare minimum. Generally a gateway, dns servers(s) and a possibly a few other DHCP Options will be specified.
This is a very very basic description of what happens and what is possible with DHCP, you should really read the DHCP RFC for more detail on how it works.
ARP is a protocol that is designed to translate IP address to MAC addresses for Layer 2 communication and isn't involved in the handing out of IP addresses any more than DNS is involved in loading a web page.