What 3 lenses should i always have for my nikon d5100?

I recently bought a Nikon d5100.
and i want to know what 3 lenses i need: to take pics of
Mtns.
people Modeling up close.
and a close shots of bugs… A really close shot. Making them a lot unique looking up close.

Gain some experience with the 18-55mm before you start getting exotic - you'd be surprised how versatile it is.

I agree with Andrew. The D5100 is so much more than a point & shoot camera that you would be doing yourself a disservice to get a bunch of niche lenses before you learn to use the camera to it's full capabilities. You need to complete your undergraduate work before going for your masters degree. I've been using (D)SLR's for 30 years, and still learn new stuff all the time. My first camera was a Canon T70, and the only lens I had for the first 5 or 6 years was a 50mm. If you just have to buy an upgrade lens, get the Nikon 35mm f1.8.It will do portaits, close ups, landscapes, and it is a very sharp lens that doesn't cost a fortune. Don't get caught up in the "My lens is longer than your lens" game. The first thing you should buy before you get any more lenses is a book dedicated to your camera, such as the David Busch books.

You didn mention your budget so i'm suggesting the best (not considering the price)

for model shoots and people… 85mm 1.4g
for close ups… 105mm f2.8 vr
for motion… 70-200 f2.8 vr

however on a tighter budget a possible list would be:

for motion as well as potraits… 85mm 1.8g and a telephoto(55-200 or 70-300)
for macro you need a dedicated macro lens.85mm f3.5 vr

For mountains, portraits, and most anything else, the 18-55mm will do. For making small bugs look huge, you will need a macro lens.

I'm glad you say "3 lenses", because there's a rationale for getting 4, or 10, or whatever, but you know to prioritize.
Okay, you have a nice wide to slight tele zoom. An even wider wide would be among the things to consider when that number of total lenses exceeds 3. Unless you want to get the 16-85mm AF-S VR and retire the 18-55mm so that it doesn't count. Maybe later.
Close up of bugs means a macro lens. There are several to choose from. Alas, live insects disfavors shorter focal lengths which have too little working distance. The Micro Nikkor 105mm AF-S VR is reputed to be a great one. It is expensive.
Then there's a telephoto. Many to choose from. Most economical is the 55-200mm AF-S VR. The 55-300mm AF-S VR is a little better. 70-300mm AF-S VR even better. There are even better lenses, with a big jump in weight and price. There's a cheap non-AF-S, non-VR 70-300mm that you should avoid. Third party lenses? Tamron SP 70-300mm f/4-5.6 Di VC USD (not their cheap 70-300mm) is worth considering. Most others, probably not. It is rumored that Tokina is working on a 70-200mm f4 that may be worth waiting for.

And don't be afraid to buy used lenses. There are some great used lenses at great prices to be had at B&H and elsewhere.

An all around lens for mountains, a good short prime for people and also other things and a macro for close ups and portraits as well.

Nikon 18-105mm
Nikon 50mm f/1.8G or 50mm f/1.4G
Nikon 40mm f/2.8G Macro

Doesn't get better than that for a low price.

Here's a Lens Buying Guide - http://www.smashingcamera.com/lens-buying-guide/