Try Ikea.
http://www.ikea.com/us/en/catalog/products/00162648/#/30216739

For basic bedding I always go for solids. I would suggest cream colored sheets and a duvet (or better yet just a topper) in anyone of the colors on the rug (dark brown or the blue). Pillowcases in a bright contrasting orange will balance out the room nicely. You can bring some additional pattern with a coordinating decorative throw or extra pillow.

Ikea or Home Goods has a good selection of solids for kids.

wool army blanket, in dark grey!

The easy thing to do is a solid color duvet cover -- plenty of them out there at your price point. You could choose a deep red, if you wanted color, or a peacock blue/teal.

If you want to mix patterns, make sure it has a different scale pattern (larger or smaller), and I'd stick to the same color family (chosen for pattern here, not price)

here are some options:
http://www.google.com/imgres?hl=en&sa=X&rls=com.microsoft:en-us:IE-SearchBox&rlz=1I7ADRA_enUS416&biw=1920&bih=759&tbm=isch&prmd=imvns&tbnid=KR5KurgJ2NUCeM:&imgrefurl=http://www.laredoute.com/products/beige-gingham-duvet-cover-and-pillowcase/706511/324216428.aspx&docid=mDxmNqcR2SiBMM&imgurl=http://media.laredoute.com/400by400/324216428_PR_1.jpg&w=400&h=400&ei=T29aT6S_JsmP0gGKpNjeDw&zoom=1&iact=hc&vpx=1546&vpy=204&dur=1147&hovh=225&hovw=225&tx=141&ty=133&sig=103372446536063060056&page=2&tbnh=137&tbnw=137&start=32&ndsp=52&ved=1t:429,r:19,s:32

http://www.google.com/imgres?hl=en&sa=X&rls=com.microsoft:en-us:IE-SearchBox&rlz=1I7ADRA_enUS416&biw=1920&bih=759&tbm=isch&prmd=imvns&tbnid=CH8euZ7xxTZMAM:&imgrefurl=http://www.houzz.com/photos/bedroom-products/p/4320&docid=DO3VfmrwY2DRAM&imgurl=http://st.houzz.com/simages/253752_0_3-3357-modern-duvet-covers.jpg&w=300&h=320&ei=T29aT6S_JsmP0gGKpNjeDw&zoom=1&iact=hc&vpx=932&vpy=239&dur=993&hovh=232&hovw=217&tx=86&ty=173&sig=103372446536063060056&page=1&tbnh=148&tbnw=139&start=0&ndsp=32&ved=1t:429,r:5,s:0

http://www.laredoute.com/products/clarisse-beige-polka-dot-print-duvet-cover/706511/324205668.aspx

its for teens/ kids I would be bold and use a kelly green or canary yellow to keep it nuetral and fun. The you could mix in so lapis accessories such as throw pillow with a little lapis or even a throw with lapis in it at the end of the bed just to bring the color from off the floor. Bedding will depend on where you are located. If it's typically warm dont spend the money on a duvet, just get a coverlet and some blankets. For the sheets just go simple with white or off white.

If you have an Ikea near you I would look that the Vila (for sheets) in the tan and Tanja Brodyr for the duvet cover, if you want something shiny and more along those lines. If you want something toned down go with the dvala and malou respectively. They are all in the tans...but you can probably find some great accent pillows for the blue color. The Vila sheet sets are 29.99 (less for twins), dvala are 24.99(less for twin), Malou duvet set for a twin is 29.99 and the tanja brodyr is 34.99 for a twin.

http://www.ikea.com/us/en/catalog/products/60083610/ This is one of the Malou...

http://www.overstock.com/Bedding-Bath/Natural-Linen-Duvet-Cover-Set-with-Bedskirt/2066644/product.html

They don't have twin but maybe you could use full? Or check out Lands End, some nice ones there. And I'd bring in some of that blue somewhere else, i think, either pillows or accents/art.

As some one who likes to pull the covers up under their chin and who has had experience with wool army blankets I give a resounding noooooo to that idea. Target has some interesting grey options in both quilts and duvet covers that might work with the rug though.

leopard print, I hate coordinated stuff!

I'd go solid, in either beige or chocolate brown, and use interesting patterns for sheets and pillows. That rug is awesome - I'd want it to be the focal point.

Bed Bath and Beyond has lots of color choices and great prices:

http://www.bedbathandbeyond.com/stylePage.asp?RN=26

LL Bean has good quality bedding at very reasonable prices. Here is a cotton comforter that comes in a variety of solids that would work with that rug and is within your budget.
http://www.llbean.com/llb/shop/57169?feat=1158-GN2

Go with cream sheets and blue or brown accessories. I don't know your wall colors you have to count that too.

It's a kids room so I would probably use a different color for each bed...Using the colors from the rug-- However I would not use white or ivory..

I would go for a solid white duvet with a bold border stripe, like this style of duvet cover, which comes in lots of different colors: http://www.pbteen.com/products/suite-organic-duvet-cover-and-sham/?cm_src=OLDLINK&pkey=gthmgrlhai

For a prints there's:

Garnet Hill Everyday Percale in driftwood:
http://www.garnethill.com/everyday-percale-bedding/bedding-home/sheets/patterned-sheets/131927?listIndex=2

or

Company Store Shelby Percale (I like the blue, but the brown might work too)
http://www.thecompanystore.com/Shelby-Percale-Bedding/e5f7-ps,default,pd.html?start=1&cgid=sheets-pattern

I'm with Herrington44 -- kids like to each have their own colour. I'd do plain duvets (one each in teal, plum, grey and navy), sheets all the same colour to save headaches, and one distinctive, fun throw pillow for each bed.

Nice rug.

I would go for tan or khaki sheets (this color is more forgiving than cream if you're using the bedding for kids and teens - probably a pet or two in the mix, right?) and that deep teal color in the rug for the bedspreads or coverlets. I'd go very simple in kid-friendly fabrics.

I don't know if I'd even mess with throw pillows for a kids room, because most pillows I've seen are "spot clean" only, and you know kids are going to grab them with grubby, sticky little hands - not to mention sneaking a snack into bed. So....I'd be practical. A few stuffed animals might be cute. Kids and teens love those brown and white sock monkeys,and they would adorable on the beds.

Target might be a good source. Also, check at www.overstock.com. They have a good selection of sheets and bedding, at reasonable prices. And for kids, you want reasonable!

Throw around a big, dark basket or two to hold games, and knick-knacks and keep them out of the way.

Use canvas dropcloths for coverlets and let each child stencil his/her name boldly (hugely oversized) in a color from the rug. Way cheap at Lowe's!

For cheap duvet covers...I buy two set of sheets and sew the flat sheet together, size of duvet cover and I will have two fitted sheets and pillow cases. Can sew sheets of different color together two, look fun when turned open halfway.
Unless you buy Ikea who has cheap duvet covers...but sometimes the fabric is not soft enough for me.

Target has inexpensive sheets and duvet sets in solid colors.

Remember awhile back there was an article about rug patterns than make people sick? THIS IS THAT RUG!!!

Agree go tan or something like that for sheets. Also would go different solid colors for the duvets or quilts, or at least two or better yet three in the room. I would paint the walls one of the 4 colors of the rug (you can go darker for the off white, and a bit lighter for the other colors if you want), and then use the other colors for the quilts. You can also mix in another color for a quilt - I like the idea of plum as suggested above. Or a rust, or the right green, one that complements the teal well.

Denim. That *Is* a blue stripe in there, yes?

nothing coordinates well w/that rug.

I would actually say something in the red, or red-orange family. Use the blue, and go it's complimentary. It should be fun! then go with the tan sheets, and maybe some pillows that match the blue/teal.

Our designer secrets, when designing your room use the 60-30-10 rule - to select colors. Your wall color would be the 60%, 30% for upholstery and fabrics and 10% accent color. In re: pattern - every room needs 3. A geometric - like your great rug, a solid and an organic (flowers, leaves, doggies, whatever).
Someone noted above that the pattern sizes should be different - we agree. So the bedding could be solid or organic, either will work.

My first thought is use one of the solid colors, the burgundy or blue, in a solid color for comforter, the other color for the sheet, with both colors in accessories, pillows and such.

Adding another color, unless it's just as "pop" accents, I think would be overwhelming.

Solid color bedding is pretty cheap. Check the usual budget stores. Duvet covers are very easy to make with sheets.

Hope that helps.