Forget the shotgun. It is not needed. Also forget the bolt action rifle. The carbine will get you to 300 yards with magnified optics. While you can shoot targets with irons, you will not see a bad guy in poor weather and light at distance.
The minimum required is a rifle. A rifle and handgun is a good combination. Optics alone will cost up to half the budget. So, I would do the following:
1) Glock 19 for around $350. Add night sights ($100). Buy two spare magazines ($50). Buy 100 rounds of FMJ ammo to make sure the gun functions ($25). Buy 50 rounds of JHP ammo ($30). Buy a cheap holster and mag holder ($100).
Total: $655+tax, round to $675
2) Bravo Company AR-15 $1,100.
Add three Magpul P-Mags (around $50).
Add a sling ($50).
You'll have to use iron sights. Adding Aimpoints (with magnifier and mount will cost around $900) or a variable optic (SWFA SS 1-4x with quick detach mount: $1,000).
3) Buy one Streamlight TLR-1 ($100). Buy a box of batteries off the 'net for $20. The light can be mounted on the Glock or on one of the rifle rails.
Total: $1,995
If you go with only a rifle, then find a decent AR-15. You need some sort of illuminated optic (again, it's about target location and identification in poor light and conditions). A sling is required, as is a stack of magazines. Some sort of belt holder for one or two magazines would be helpful.
You could go with a non-illuminated optic*, but your performance will degrade as the light decreases. There are several lower cost variables in the $500-800 range (excluding mounts) that may be adequate. If you are not sure, buy good iron sights and add an Aimpoint optic. You need the irons in case the battery goes dead from use or cold.
I used a Leupold VX-3 1.5-5x with circle-dot reticle out to 300. A 100 yard zero made it very easy to use. However, the 1.5x really messed with my times under 25 yards.
I have found that a basic AR-15 with good ammo will score hits on demand out to 300 yards. This should be good enough for any home defense situation you may encounter. If someone is shooting at your house from 1,000 yards, you're going to have a very difficult time finding him--especially if you don't have the right training or experience.
Here are some lower cost variable optics to look into:
Vortex Viper PST
http://swfa.com/Vortex-1-4x24-Viper-PST-30mm-Rifle-Scope-P44570.aspx
Capped Turrets:
http://www.vortexoptics.com/product...-with-tmcq-moa-reticle-capped-turrets/reticle
Burris
http://swfa.com/Burris-1-4x24-Xtreme-Tactical-XTR-30mm-Rifle-Scope-P45510.aspx
Leupold VX R Patrol
http://swfa.com/Leupold-125-4x20-VX-R-Patrol-Tactical-30mm-Riflescope-P49444.aspx
Weaver 1-5x24 Tactical 30mm Rifle Scope
http://swfa.com/Weaver-1-5x24-Tactical-30mm-Rifle-Scope-P49447.aspx
Trijicon Accupoint
http://www.trijicon.com/na_en/products/product2.php?id=AccuPoint&mid=1-4x24
* Take a look at the Leupold Hog Pig Plex. It's a 1.25-4x optic for under $300. It has a decent reticle and 300 to 400 yard shots should be easy.
True 1.0x up to 1.1x is better for CQB and defense. However, prices for scopes with that setting tend to be more expensive.