I don't actually have any idea what the f/4-5.6G means.

The "ƒ/4-5.6" part is the maximum aperture of the lens. It's a variable maximum aperture, meaning the f-number is not constant through the lens's zoom range.

The G part means it's a "G"-type lens, which is Nikon's designation for a lens without an aperture control ring. That means that the camera must have aperture control. It's can't be done by turning an aperture ring like on older or fully manual lenses.

See also, the Nikon answer to What do all those cryptic number and letter codes in a lens name mean?

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In general, when searching for lenses for your camera, I suggest that you refer to:

1. Nikon's product page for the D3300. Under the Tech Specs, it lists (bold emphasis mine):

   Lens Compatibility at a Glance

   • AF-S Lens Required for Autofocus

   Compatible Lenses

   • AF NIKKOR for F3AF not supported.
   • AF-S, AF-I: All Functions Supported.
   • AI-P NIKKOR: All Functions supported except autofocus and 3D Color Matrix Metering II.
   • Electronic rangefinder can be used if Maximum Aperture is f/5.6 or Faster.
   • IX NIKKOR lenses cannot be used.
   • Non-CPU: Autofocus not supported. Can be used in mode M, but exposure meter does not function.
   • Other AF NIKKOR: All Functions Supported Except autofocus and 3D Color Matrix Metering II.
   • Type G or D AF NIKKOR: All Functions Supported Except Autofocus.
   • Type D PC NIKKOR: All Functions Supported Except Autofocus and some Shooting Modes.

   Note that the product page is probably older than Nikon's latest lens technology, AF-P lenses, such as the AF-P DX 70-300mm ƒ/4.5-6.3G ED VR. They will autofocus on a D3300, but only with a firmware update (available from Nikon). At Nikon's support page, Can I use an AF-P lens with my DSLR camera?

   The following cameras are also fully compatible with AF-P lenses when the firmware is updated to the following versions:
   ...
   D3300 (firmware C: Ver. 1.01 or later)

   Restriction for D5300 and D3300 only

   As D5300 and D3300 firmware updates do not include the "Manual focus ring in AF mode" Custom Setting included with the D5500 firmware update, the focus ring can always be used for manual focus when the camera is in autofocus mode.

2. Nikon's support article, Which lenses can be used with a DSLR camera which has no built-in autofocus motor?

   The following digital SLR cameras do not have an autofocus motor built-in the camera body:

   D5600, D5500, D5300, D5200, D5100, D5000, D3400, D3300, D3200, D3100, D3000, D60, D40X, and D40.

   The cameras listed above use NIKKOR AF-S, AF-I or AF-P type lenses which have the autofocus motor built-in to the lens itself. This negates the need for the motor in the camera body allowing the camera to be smaller by design.

3. Wikipedia's List of Nikon F-mount lenses with integrated autofocus motor. Currently, I believe all the lenses on that page can autofocus with your camera body.