Netgear Nighthawk AC1750 wireless router and AC600 Wifi USB Adapter

> [...] Nighthawk AC1750 wireless router and an AC600 Wifi USB Adapter

   "AC1750" or "AC600" is a speed, not a model number.  Ever helpful,
Netgear product packaging emphasizes speeds like "ACxxxx", but that's
not the model identifier.  Look at the product label.

> [...] Do I need to uninstall the old router and adapter before
> installing the new Netgear products? [...]

   Installing two USB wireless adapters at once is probably unwise.
They'd be very close to each other, which I'd expect to cause
interference, and you'd probably need to tell Windows to use only one at
a time.  Too much potential confusion/trouble.

   Presumably, your router is connected to some kind of modem, which has
only one LAN port, so trying to connect both routers to it at the same
time would be impractical.  And unwise: the router is a sort of master
controller for your LAN, and a network can't serve two masters.

   You might want to start by replacing the USB adapter, and getting
that to work with the old router.  Then, when everyone's happy, swap the
routers.
 
> How will I know whether my network is secure once I install the new
> router and adapter?

   How do you know that your network is secure now, _before_ you start
swapping hardware?  Set an "admin" password for the new router (other
than the default, "password").  Configure the wireless credentials
(SSID, passphrase) on the new router to match those on the old router.

   Visit http://netgear.com/support , put in your (actual) model
number(s), and look for Documentation.  Get the User Manual(s).  Read.
If mysteries remain, then ask specific questions.