Assistance with Modem & Router Setup

> [...] I'm using a Docsis 3.0 modem provided by Comcast [...]

   An actual model number might convey more useful information.

> I have my Playstation 4 hardwired into the Docsis [...]

   "DOCSIS 3.0" is not a maker/model identifier.

      https://en.wikipedia.org/wiki/DOCSIS

   Presumably, the Netgear WNR1000v2 (WAN/Internet port) is also
connected to (a LAN port on) the Comcast-supplied gizmo, which would
suggest that, whatever it is, it's a modem+router.  I'll assume that.

> I also have several mobile devices that connect to the Docsis.

   So, those are wireless connections?  (And the Comcast-supplied gizmo
would seem to be a modem+(wireless)router.)

> I'm still at a loss of understanding why any of the devices connected
> directly to the Docsis modem cannot see both of my computers over the
> network.

   That's easy.

> Docsis Modem:
> Default Gateway 10.0.0.x
> Subnet Mask 255.255.255.0
> DHCP enabled

   A default gateway has a definite IP address, not a ".x" address.
Devices connected to that modem+router may be assigned addresses in the
10.0.0.* range, but their default gateway should be the gizmo itself,
which, I'd guess, is at 10.0.0.1.

> Netgear WNR1000v2 Wireless-N 150 Router
> 192.168.1.x
> Subnet Mask 255.255.255.0
> Starting IP range of 192.168.1.x to 254

   Again, I'd guess that the WNR1000v2 has a (LAN) IP address of
192.168.1.1, which would be the default gateway for devices which are
connected to it.  And those devices get addresses in the 192.168.1.*
range.

> If I access the Docsis modem's configuration page, it is assigning my
> Netgear router an IP address within the 10.0.0.x range.

   Just as it does for everything which is connected to it.

> Is there anything that I can do to make all the devices on my network
> see each other aside from having them all go through the router (which
> would require me to buy a new one with more ports or by using a switch)?

   Perhaps, but fair warning: Adding a switch to provide more ports to
the WNR1000v2 network would be the easy way, and should be pretty cheap.

   The problem, as things are currently configured, is that devices on
the modem+router sub-network (10.0.0.*) have no idea how to reach other
devices which are on the WNR1000v2 sub-network (192.168.1.*).  Normally,
messages for addresses which are on a foreign network (not 10.0.0.*) are
sent out the modem+router's WAN port (to your ISP).  (In this case,
foreign addresses like 192.168.1.* would normally be considered private
and non-routable, so they'd probably be discarded rather then sent off
to annoy your ISP.)  In any case, the modem+router has no reason to send
such messages to the WNR1000v2 for further action, which is what would
need to happen.

   There are different ways to deal with this situation.

   1. Buy a network switch, use it to expand the ports on the
   modem+router, and forget about the WNR1000v2.  If the wireless
   features of the WNR1000v2 beat those of the modem+router, then this
   option might be unattractive.

   2. Buy a network switch, use it to expand the ports on the
   WNR1000v2, and connect nothing to the modem+router except the
   WNR1000v2 (WAN/Internet port).  That includes any wireless devices.

   2a. Configure the modem+router as a modem (only).  This is often
   called "bridge mode".  Stacking one router after another can work in
   simple cases, but it complicates features like port forwarding. 
   Using only one router simplifies management of such things.

   3. If the modem+router allows it, then add a static route on the
   modem+router, which would tell it to send messages for 192.168.1.*
   addresses to the WNR1000v2 (WAN/Internet port).  To make that work,
   you'd also need to fix the IP address of the WNR1000v2 WAN/Internet
   port, so that the address in the static route definition remains
   valid.  (You could assign a static address, like, say, 10.0.0.251, to
   the WNR1000v2 WAN/Internet port, and then take steps to ensure that
   that address is not in the DHCP pool on the modem+router.  Or, if the
   modem+router allows it, you could reserve a dynamic address for the
   WNR1000v2 WAN/Internet port.)

   See how attractive adding a switch looks?

   4. Configure the WNR1000v2 as a wireless access point instead of a
   full-function router, so that devices connected to it would be
   members of the modem+router subnet (10.0.0.*).  If the WNR1000v2 as a
   full-function router has features which the router part of the
   modem+router lacks, then this option might be unattractive.

   There may be more possibilities, but that's decent start.  Questions?