There's nothing worse to getting home, turning on the radio, and being unable to find a clear frequency that isn't full of a bunch of meaningless QSOs where somehow everyone is 599 and no one has anything to say. Are there any designated "contest-free zones", or other ways I might be able to enjoy amateur radio without contests?
Most countries follow the band plans that disallow contesting on WARC frequencies, which include three narrow slices of shortwave frequencies at 30M, 17M, and 12M. If you stick to these frequencies you will rarely, if ever, be bothered by contests.
It's a small slice of spectrum though, so you might simply consider doing what some amateurs do and simply note the days the contests are operating, and plan activities other than amateur radio on those days. Given that most contests are short duration, occur mostly on weekends, and don't occur more than a few times a year, it's not unreasonable to simply avoid affected bands on affected days.
One of my approaches to avoiding contests is operating on the opposite end of the band. During a big contest weekend it can be tough to avoid, when calling CQ, and someone comes back with ID and contest requirement, so the best thing to do is find the extremities of the band. Usually it's the upper end of the band, like for instance on 20 meters, I'll just call CQ in the 14.330 MHz to 14.350 MHz area. It's generally quiet there, and I have found that most call backs are other hams looking to rag chew. Like an earlier answer pointed out, just avoid the traffic by going around it.
Apart from the WARC bands, you could possibly try CB radio. I know in the US it usually has a negative stigma attached to it; but a lot of my local HAMs are similar to you, as in they hate contests, so they go on CB channel 38 when they can't get through on the other HF bands.
10m doesn't get used very much here, and even if it does, we get horrible QRM, so 11m (CB) is the last resort.
However, if you want to be a 'reckless' ham, you could always transmit voice over the CW and data frequencies. People might hate you for it, but It's not illegal, at least not here (UK).
I strongly disagree with such kind of well known, instant answers (myths) that contests are short and rare, use WARC band instead, you can participate them with small power and antennas anytime to make more shiny DX's, and all because they are arrogant in their senselessness despite of any amount of goodwill. I know that I am in the minority with this opinion but I feel it important to express, that's why I spend time to note them. Some competing can be fun and useful but contesting on "higher levels" is a wrong, instinctive, and underregulated approach to the radio amateur principles, namely the learning, building and experimenting. Ham radio is not a sport, it is intended to be much more competent. There can be compromises but in this topic you asked how to avoid them, and it signals something... Even some hams with very big antenna farms declare on their web pages that they are not interested in contesting but DX-ing.
Other radio amateurs have absolutely no right to send you somewhere else from our precious collective resources called ham bands that you are licensed to use to another ham band or segment. Nothing can justify it. Why? E.g. because somebody might have a radio or an antenna only for that very band, or only for the classic bands (not for WARC bands), etc., you can imagine. What you see instead? In the last years, even the IARU recommendations miss the lowest 10 kHz sections of bands those are traditionally for DX-ing. Also, the important calling frequencies, like QRP or other mode's calling frequencies (those should have an absolute priority) are completely surrounded by the recommended stretches of the contests. That's insane (regarding to the average level of discipline of ham folks).
Many (most!) weekends are full with contests, and stations on steroids, at least here. They take every space because “the contest needs it”, and you must obey. There are exceptions as everywhere, though. I respect the technical aspects but small stations are completely unable to make a single QSO many times at weekends when most of the people have a little time for the hobby. There is no place, therefore and mostly there is no partner. There are even larger pile-ups on DX's you can't penetrate since that is a contest and there are tons of kilowatts in the ether (even if tons of kilowatts is not an SI unit, hi).
There are incomplete reports in QSO's. What is in my license? Sending-receiving the RST's are obligatory just like logging the frequency. It is true at least in my country, but it is a tradition as well, anyway. What is in the contest logs? Always 599 or 59. Because “the contests need it this way”... It is only an agreement, not a necessity. And they go further: some contests got rid off of the RST's at all. But, to say something good, too: on V/U/SHF, where contesting is much more acceptable, there are real RS(T) reports. Such contests are usually much closer to the ideal ham activities including technical aspects as well.
Do not forget: if you make contest QSO's, it is a bad manner not sending at least a control log, so that could be the next thing you did not want...
Let me say again that I understand the compromises, I also take part of contests from our club stations (because those are the events having enough gravity to collect people to one place), but I'm rather displeased with the general attitude toward the small but equally important cultivators of this hobby, the wrong priorities and wrong habits. I don't think the marked answer is the real one for you.