Connectivity issues with wireless router?

The internet service running through the house is provided by Xfinity->Comcast. Up until a couple of months ago everything worked just fine, we had a modem provided by Comcast and a Dlink wireless router. At one point, we decided to switch to a faster internet plan, had to bring back the modem, and we were given a new modem that was also a wireless router, eliminating the need for the Dlink router.
The internet speed is indeed a bit faster, but we have been experiencing some very frustrating connectivity loss issues. In the house we use 1 iPhone, 1 iPad, and an Asus UX31 ultra slim PC.
Most issues appear on the PC, in the sense that when it is opened after a period of being shut down and an internet explorer is opened (usually Firefox or Chrome), the web pages can't be loaded (sometimes a "server not found" message pops up) for approximately 1 minute. During this time the PC shows that it is connected to the wireless connection in the house and good signal strength and that there's internet access, however websites can't be loaded. Afterwards, throughout the online session, the internet "drops" for minutes, out of the blue, and then recovers normally as if nothing has happened (without any interference on my end). During this connectivity issue, if I try disconnection from the wireless connection, and then reconnecting, it will tell me the first 5 times that Windows was unable to connect to the wireless connection, and then after another couple of tries it just works.

Recently we have noticed that once the PC is opened and an internet explorer is opened, the internet speed greatly reduces if not stops working altogether on the Apple products, for the set 1-2 minutes it takes the connection to start working properly on the PC.

When the new modem was set up all settings were made default by the setup wizard.
No changes were made to the PC's firewall.

On some laptops going into sleep mode or hibernation mode causes the laptop to turn off the network card. And when the system "wakes up," the network card has a hard time getting back into the game.

Try this for the next few times you use your laptop: When finished using your laptop -- totally shut it down so that it powers off. Yes, it's a pain, but you need to see if your laptop behaves when you power it back on the next time. I bet it comes up without a network problem.

So try that test a few days/times. Then, if powering it off/on works fine, then we can give you suggestions on sleep or hibernation modes.