Encryption algorithms are constantly being tested and broken
Oh, there's no doubt that cryptographers are continually creating and breaking new ciphers. But there are well-proven ciphers that have stood unbroken for decades. DES, for example, although the key size is now too small, has stood strong for 40 years, and in fact if you don't mind applying it three times is still proof against anyone in the world, as long as you keep the keys safe.
I have little doubt that AES will do just as well, and its much larger key size means that we won't face brute force becoming tractable, not unless actual breaks in the algorithm are discovered.
From the NSA and other governmental entities deliberately weakening the tools we use to encrypt
The fundamental algorithms, like AES, are relied upon by governmental entities, which the NSA is tasked with securing. We do have evidence that the NSA has been working in various ways to compromise implementations, but they can't really do that with ciphers, because if you modify a cipher, it won't pass the standard test vectors. The NSA can try to introduce things like side channel attacks, but if the NSA is actively monitoring your machine while you encrypt and decrypt (which is needed to exploit a side channel), you should just give up and flee to Russia now.
Really, if you use standard, widely-trusted cryptographic algorithms and are careful with how you generate and manage your keys, no one is going to get your data. Not even the NSA. Generation can be a little tricky to get right, but there are lots of good solutions. Probably the best is to randomly (e.g. with diceware) select a string of a dozen or so dictionary words, then use them with a standard key derivation function to produce a key, then use that to encrypt your data.
If that sounds complicated... it's really not. "gnupg -c" does a fine job. Run that on your files and upload them to the cloud. Since you're apparently happy with the security of your home machines (ha!), you can store a copy of the passphrase on the machine that does the backups, so you can automate everything. So that you can recover your data in the event of a fire, write down the passphrase on a piece of paper and store it somewhere offsite and secure. Give it to a friend or family member that you trust, or put it in a bank safe deposit box.
Encryption, cloud storage and an offsite copy of the decryption key are the way to go here.
If you want a nicely-automated version of all of that, you might want to look into the services provided by leastauthority.com.