virtual IP will be bound to an Active role in the HSRP - depending on the priority the one elected as ACTIVE will hold the IP you see that information
show standby brief show IP arp (show the virtual IP and mac address of virtual IP)
The point of HSRP is the redundancy effect. You could point your devices to the default GW of the 10.0.0.1 IP address but if that router fails you will lose your GW and the devices wont know where to go. If you point it to the 10.0.0.3 (VIP address), then if R1 fails you have R2 as a backup also using that IP. You point the clients to the VIP or the physical IP. I could possibly see a scenario where you would point a client(s) to the physical IP if it had to go that way for a requirement...but like I said you lose the redundancy.
Also, I wouldn't call the physical IP of R1 an "Active" IP, while technically it is active and functional, the terminology mixed with the HSRP configuration could be misinterpreted.
Don't forget you can also configure multiple secondary IP addresses on an interface as well.
Hope that helps
"With a Hot Standby Routing Protocol configuration, does the active router have 2 active IP's on one interface at same time (The normal= 10.0.0.1, and the VIP= 10.0.0.3)?"
If by "active router" you mean the one which is actively being the HSRP IP, as noted by the other posters, the answer is yes.
As other routers can be not concurrently be the same "active" HSRP IP, on those only their physical IP will be "active", i.e. they will not, for example, respond to a ping to the HSRP IP.
BTW, other FHRPs work a bit differently. For example, if using GLBP, all the routers supporting the same IP might do so at the same time. I.e. if you ping the GLBP IP, you'll get a response from from one of the GLBP routers but which one respond might differ based on which host pinged the GLBP IP. (All those routers, though, will respond to their own physical interface IP.)
Also with later variants of HSRP, you can have multiple HSRP IPs. Depending how you define priorities, and which router "wins" the election to "actively" be a particular HSRP IP, one router might be the one responding to multiple HSRP IPs on it, or, again due to how the HSRP IPs are currently actively hosted/supported, different routers might respond to a specific HSRP IP.