Yes it is possible to do ski trip on a reasonable budget, it will never be cheap though unless you can crash at someones place and you already own your own equipment.
I live in Colorado, so I go skiing up there all the time and I am a part time ski instructor, so I may get a little long
I would first suggest stopping by a king super or city market on the way up to ski area, you can normally find cheaper tickets there for the ski areas. Also rentals are a lot of time cheaper down in Denver then getting them up at the ski area. Also if you plan on returning at all, buying a season pass is a great way to go to save money, normally you can pay for it in about 4 to 7 days depending on when you buy them. The earlier you buy them the cheaper it is.
Also if you can I would look into other ski areas. Breck is one of the most popular ski areas in all of Colorado and presidents day weekend is one of the worst weekends, think X-Mas in Magic Kingdom and you get the idea. I would look into Keystone, Copper, A-Basin or Loveland. They are all in the same county, so not hard to drive around to or take the bus. (They are the Universal Studies or SeaWorld in Orlando) But the lines are bad and I mean bad on that weekend. If I am not working, I will be skiing at A-Basin that weekend since it is smaller and not as popular. Also it is normally easier to find deals for the smaller ski areas like Loveland and A-basin.
Also it is worth paying for at least one ski lessons, I may be biased (as a ski instructor) but I see so many people get hurt and then ruin the whole rest of their ski vacation because they were trying to save money by not taking a lesson. So in the end of the day save your self pain and misery and book one day of lessons. Also on a side note Breck lessons for kids includes at lot of stuff including food and rentals. Lessons are also cheaper at smaller ski areas than they are at Breck too and just as good. If you guys are thinking at all between skiing and snowboarding, skiing is easier to start with so I normally suggest that for shorter stays, but if you plan on doing this a lot, snowboarding is easier to progress after the first few days.
Also always grab the Summit Daily newspaper, it is free and its has great coupons in it and it shows you what the restaurant deals are in the area that day. And when ever I am in Breck I take advantage of the happy hours from 4 to 6pm at the high end restaurants, you can get the best meals at great prices. My favs are Relish, Briar Rose, and Twist in Breck.
In Frisco a town right next to Breck (that you will drive through) and there is a
Walmart and Safeway for buying all the food and buy huge packs of hand warmers. Do not waste money on foot warmers, they do not work because they need air.
Also there is a Outlets in Silverthorne, a great places to find good deals.
And of course other people already mentioned pack your lunches, it was I do too
Things that have nothing to do with budget, but good ski tips
Bring good ski socks, pull them all the way up make sure nothing is bunching in boots. So do not stuff pants and long underwear in the boots, they cause pressure points. Pull the pants up and over the boots.
Also this another thing that people do not think about, altitude sickness is a real thing that normally one or two of my kids in everyone of my lessons get. They normally just arrived the night before from sea level and did not get a lot of sleep the night before. So please drink lots of water and if you can the first day you are here do not ski, do tubing instead or dog sledding (there is a fun one in Breck I love) and give your body one day to get use to the high altitude because if you or your kid gets sick they do not give you a refund with the ski lessons.
Always wear a helmet, it does not matter how well you ski, I have seen so many people get hit by another person. So even though I use to be a ski racer I always wear it just in case

Another good tip, stop on the side of slope so you do not get hit by people.
Have two pairs of gloves for each person so when one is wet, you can switch out with the other one and of course wear layers. Also bring sunblock and use it, you will be amazed at how easily you can get a sunburn here during winter.
Okay I am soo sorry for going so long. But you will have so much fun, I love skiing and it is something your kids will just love. I probably missed things, I am in summer mode not skiing.