Do I have to get a specific setup to mine a specific cryptocurrency, or can I use a generic setup to mine most/all of them?

There are many options for cryptocurrency mining.

Some algorithms can still be run more or less 'effectively' on CPUs (eg, Cryptonight), others work best on GPUs (Ethereum, Zcash, Vertcoin), and still others are the domain of custom ASICs (Bitcoin, Litecoin). But besides having the hardware, there are other steps to take to get started with mining.

• In the early days of Bitcoin and some other cryptocurrencies, you could effectively solo-mine the algorithms. That meant downloading (or even compiling) the wallet for a particular coin and the correct mining software. Then configure the mining software to join the cryptocurrency network of your choosing, and dedicate your CPU/GPU/ASIC to the task of running calculations. The hope was to find a valid block solution before anyone else. Each time a block is found, the calculations restart, so having hardware that can search potential solutions more quickly is beneficial.
• These days, a lot of people forego running the wallet software. It takes up disk space, network bandwidth, and isn't even required for mining. Just downloading the full Bitcoin blockchain currently requires over 45GB of disk space, and it can take a while to get synced up. There are websites that take care of that part of things, assuming you trust the host.

In theory, over time the law of averages comes into play. If you provide one percent of the total computational power for a coin, you should typically find one percent of all blocks. But as Bitcoin and its descendants increased in popularity, difficulty shot up, and eventually solo-mining became an impractical endeavor. When you're only able to provide 0.00001 percent of the mining power, and that value keeps decreasing over time, your chance of finding a valid block solution becomes effectively zero.