The simplest way to clean out your cache and log files is with a utility such as Onyx from Titanium's Software (donation requested). After launching Onyx, go to the Cleaning tab. For starters, delete only items in the User and Logs sections.
It's possible that a bug, such as a "runaway" log file, may be causing your drive to fill up much faster than it should. If your available free space returns to near zero shortly after you clear out the files, a bug of this type may be the culprit; if your Mac has this symptom, check the Web for possible solutions.
Check CPU usage
If your Mac's Central Processing Unit (CPU) is overwhelmed by an app, everything on your system may slow down.
Launch Activity Monitor and select My Processes from the pop-up menu at the top of the window. Next, click the % CPU column to sort by that criterion.
If an app consistently remains at or near the top of the % CPU list--and especially if it accounts for an atypically high percentage of the CPU's workload (probably anything over 50 percent, and certainly anything over 80 percent)--that app may be gumming up the works. To find out, select the app and click the Quit Process button.
By far, the most likely source of trouble is Apple's Safari (and more specifically, a webpage that uses Flash). If webpages are loading extremely slowly, and if the % CPU for Safari and/or Safari Web Content remains high, it's time to take action.
To reduce the odds that a Safari slowdown will recur, minimize the number of webpages you keep open at one time. You might also try using Google's Chrome: If one tab misbehaves in Chrome, you can use Activity Monitor to zap it without bringing down the whole app.
Check memory usage
Macs using hard drives depend on a combination of physical memory and virtual memory to get things done. Physical memory is faster. The more heavily your Mac has to depend on virtual memory, the slower it will perform. Virtual memory also creates swapfiles that increase in size over time. (To find these, select Go > Go to Folder in the Finder, type in /var/vm, and click Go.) Swapfiles can contribute to a system slowdown by using up disk space.
Quit apps: To improve matters, quit apps that you aren't currently using. Then restart your Mac.
Check memory usage with Activity Monitor: As before, to check for memory usage problems, launch Activity Monitor. Look under the column headers Real Mem and Virtual Mem. If an app is using a disproportionate amount of Real Mem and Virtual Mem, you can quit it by selecting it in the list and clicking Quit Process.
I generally focus on the System Memory statistics at the bottom of Activity Monitor. (Click the System Memory tab to see these.) If the 'Page outs' and 'Swap used' values are high (over 2GB, as a rough approximation) and the amount of free memory in the Free listing approaches zero, insufficient memory is probably contributing to your slowdown.
Deal with persistent problems: If speed and memory problems remain, or soon reappear, one of two things may be responsible. First, the problem may be a "memory leak"--a bug that causes a particular app to use excessive amounts of memory. Often a Web search will confirm this situation and offer further advice. Second, your Mac may not have enough installed memory to meet your current needs.
Money is the ultimate cure for stubborn slowdowns.
Add RAM: If your Mac doesn't already have its maximum amount of memory installed, and if its memory is accessible for upgrades, adding RAM is the quickest and cheapest way to add zip to your Mac. For more information about adding RAM, see Apple's guidelines for the MacBook Pro, the MacBook, the Mac Pro, the Mac mini, and the iMac.
Get a bigger hard drive: If you continually bump up against your hard drive's space limits, consider replacing it with a larger-capacity drive or offloading some of your files to an external drive. Shifting from a conventional hard drive to an SSD drive can speed things up, too. The site Ifixit contains useful step-by-step guides to replacing the hard drive on many Mac models.
Buy a new Mac: If you can afford it--and especially if your Mac is more than three years old--your best bet might be to buy a new one. A new Mac is likely to have a faster CPU, more memory, a larger faster drive, faster ports (such as USB 3.0) and improved Wi-Fi performance. Add it all up, and it equals no more slowdowns, until the cycle repeats itself in a couple of years.