OK, how about a real answer for a change?
I live in Minnesota, so I understand your climate needs and I work in the tire industry so unlike the other people who have responded so far, I have some idea what I'm talking about.
I find it shocking that you would still be riding on the original hoops after 80k miles and 7 years. When new these cars came equipped with either Bridgestone Potenza RE92 tires (15-inch) or Michelin MXV4+ (16-inch), neither of which has a reputation for lasting much longer than 40k miles and the Bridgestone's are horrendous in snow even when new. If you still had those you would have got rid of them at the first snowflake. As I alluded to, your 2000 Avalon will have one of two tire sizes fitted: P205/65R15 92H or P205/60R16 91H. I'm not sure which size is fitted to your particular vehicle so I'll make suggestions for both. I have personal experience with all of the tires I will talk about in heat, rain and snow and ice conditions either in the real world, on the test track, or both.
Ignore the guy who said you need "all-terrain" tires. That is too ridiculous for a response.
Ignore rick and his advice about "traction ratings" too. The UTQG ratings that are given to all passenger tires in the U.S. are about as useful as t-ts on a bull. They don't tell you anything about how a tire is going to perform in the real world.
In either P205/65R15 92H or P205/60R16 91H I can recommend the BF Goodrich Traction T/A H as a competent al-season performer at an attractive price point. 60k mileage warranty on that one.
A better but more expensive choice would be the Bridgestone Turanza Serenity which is really a marvel of tire engineering. Fantastic in every way. 70k mileage warranty to boot
Michelin's Primacy MXV4 is the replacement for the MXV4+ that has been a popular Original Equipment fit tire for many years. The Primacy promises longer wear and better traction and probably has the lowest rolling resistance of the bunch (for good gas mileage). I have not driven this tire in snow yet. It has only been out for a few months so I just have not had a chance, but they should be decent enough. for the money though if it were my own car I would chose the Bridgestone's.
The Goodyear Assurance Tripletread (make sure it is the Tripletread, not the Comfortread) gets great reviews for rain and snow traction and has a very long 80k mileage warranty to boot but I think there are some supply issues with these right now. If you can find them this would be my #2 choice behind the Bridgestones. Someone else mentioned the Goodyear Eagle ResponsEdge. I would not recommend this tire to someone living in Denver. That tire is a "hihg-performance" tire for people who like to play Mario Andretti around the cloverleafs. You don't strike me as the type. The Tripletreads will be orders-of-magnitude better for the way you use your car.
If it were me and I lived in Denver I would actually invest in 2 sets of wheels and tires - one for summer and one for winter, the latter fitted with tires dedicated specifically for winter use such as the Bridgestone Blizzak WS-60. I live in Minnesota and this is what I do on my personal vehicle. A dedicated winter tire like that is just so much better in very cold temperatures, snow and ice than so-called "all-season" tires that you can not even compare them. Remember, if you read the fine print "all-season" tires are only useful in "light snow". Also all-season tires tread rubber loses elasticity (traction) at temperatures below 45-deg F so that even on a cold day without any snow or ice a winter tire will work better than an all-season tire.
The Tirerack.com is a great place to get wheels and winter tires because they are extremely price competitive and will mount and balance the tires at no charge when you buy the wheels at the same time. You can buy just tires from them too and while their tire pricing is very cheap, once you add shipping and the cost to have them mounted on your wheels locally, your nieghborhood tire shop can often beat them on price.
If you are unwilling to make the investment in two sets of tires to optimize your tires performance all year round but care about not dying in a horrible fiery crash in the next blizzard there is an alternative:
The Nokian WR.
These tires are made in Finland (the Finn's know snow and cold) and are regarded as "all-weather" tires, not "all-season". Basically they are a winter tire, complete with the severe winter service rating on the sidewall but because they don't wear out as quickly as other winter tires you can use them all-year round. I have these tires on my wife's car and they are amazing in the winter - they blow away any all-season tire - and quite good in the summer as well. After 30k miles they are only half-worn which is fantastic given how aggressive they are. Nokian tires should not be hard to find in Denver.
Irregardless of which tire you chose, I always recommend replacing all four so that you have the same traction on all four corners of the car and you can maintain a proper tire rotation schedule which will yield the best wear.
PS
The "penny check" is no good. It's a relic from the days when the big concern about tread depth was warranty claims for treadwear, not traction. The penny test shows a tire worn to 2/32" remaining tread (the legal minimum in most states) but traction begins to fall off dramatically in tires once they wear below 4/32" remaining tread depth. You need a quarter for that test, not a penny. As apoint of fact, there was a news story just today that Consumer Reports is now recommending tires be replaced at 4/32" remaining tread, not that I put much faith in their opinions most of the time, but they are right on this one.