I presume that you are connecting the phones to a VoIP phone system elsewhere, rather than to a local VoIP phone switch. If that is true, the phone provider will likely have an opinion. Be aware that their opinion serves their needs (reducing support calls), not necessarily your needs (ease and cost effectiveness).
Do you recommend the VOIP phones go into the same router as the rest of the network or two routers (1 for VOIP, 1 for all else)?
I recommend a single network. Set the phones at the desk, connect them to the main switch, and plug the PCs into the phones. A single 24 port switch will suffice, provided it has PoE. Confirm that the switch can provide maximum load to all of the ports at the same time! I have had good luck with HP ProCurve. Get redundant power supplies on the switch, and a good UPS with battery backup. Do the math to make sure you have several hours of battery life.
Plug the internet router and the servers into the same switch. You will end up with a clean, easy to manage network that will perform well.
Setting up two networks sounds cleaner, but it isn't. You end up with twice the hardware, which means twice the expense and triple the management hassle. You may need to setup VLANs and put the phones on a separate VLAN, but I doubt it will be necessary with 14 phones. You should be able to configure DHCP to allocate the phones and PCs to different pools if you are so inclined.
What kind of router or routers do you recommend to handle all this traffic?
I defer to others on this question, as I have only done local VoIP. That said, I don't think it will be that much bandwidth ( < 800 Kbps max). You do need to make sure that the router supports QoS, and ideally that you can get QoS all the way to the phone provider.
With 14 VOIP phones do we need to do anything to prioritize traffic for the phones on the router, or is not not necessary with a small deployment?
With 100Mbps to the internet and local Gigabit, you should be fine as long as you get sufficient bandwidth (~ 800Kbps) to the phone provider. Check with the phone provider to see if they require anything. They likely recommend Quality of Service (QoS), and may require it.
If you start getting bandwidth-related problems, setup QoS. The telltale sign of bandwidth issues is the calls getting tinny. You setup the phones, the switch, and the internet router, and have the internet provider configure it.