We're generally aiming for a cryptographic strength of $2^{128}$ (i.e. close to $2^{128}$ operations are required to break a scheme). This is also called a cryptographic or security strength of 128 bits. An operation is anything that can be used to test a key value or possible break. Lower than that is possible for specific instances (real time cryptography) but 128 bits of security are generally considered safe.
For which key sizes or hash functions should be used to obtain this kind of security please have a look at https://keylength.com, e.g. going into the NIST recommendations. Beware though that this only shows secure key or output sizes. It is much more likely that protocol or implementation errors will result in security issues than a smaller key size.
Normally when we show these kind of values we only consider classical computers, possibly run in parallel of course. Against full fledged quantum computers we can have a similar security value, so $2^{128}$ operations running on a quantum computer. Beware though that this is probably a lot harder to do, which is why cryptographers may still consider AES-128 secure even though it may "only" require $2^{64}$ operations of a quantum computer to break it.
A full fledged quantum computer will however make many common asymmetric primitives such as RSA, (EC)DH and ECDSA insecure. You could take a look at Post Quantum Cryptography to choose a Quantum-Secure Cryptographic algorithm.
$2^{60}$ operations has never been considered secure against all adversaries. $2^{80}$ is about the strength of SHA-1 and 2-key (128 bit key of which 112 bits are used) triple DES. That has been seen as secure in a legacy sense for a while by NIST, but it isn't considered secure anymore. The very bare minimum that should be used is 112 bit strength which for instance is offered by three key triple DES and 2048 bit RSA or DSA, but 128 bit security should always be preferred.
The SHAttered attack that broke SHA-1 collision resistance took $\sim2^{63}$ operations. Single DES with 56 bits has been broken for considerably longer (and took only 16 times less effort than 60 bit security would require). 60 bits just doesn't do at all.
