Webmaster Tools has a nice feature called "Site performance", under the Labs section. It will show a graph of average load time over the past few months. Unless your site is completely static with low traffic, it will almost certainly be in the red area so don't worry about that.
In some cases (I assume for bigger sites) it lists some of your slowest pages and some suggestions to improve page speed. Those suggestions come from Google's Page Speed tool, so your best bet is to install that, and Yahoo's YSlow tool, to help diagnose any speed issues. Search Pro Webmasters and Stack Overflow too, there are several questions on improving site speed here.
I don't think Google Analytics has anything that shows page speed. There is a "Connection Speeds" page, supposedly showing your visitors' internet speed, but for me this has always shown "Unknown" and nothing else.
In terms of usability, "machines" cannot really help you with that since it's a human reaction. Do some user testing yourself. Accessibility is a useful goal; as usual Wikipedia has plenty of information. There are accessibility checklists around too, just search for them.
For general SEO, you'll want to keep an eye on your top pages and referring keywords in Google Analytics. Use Google's Adwords tool to check if there are important search phrases you are missing out on.
GWT also has a slew of features. "Search queries" shows what your visitors are searching for, similar to GA. Here though you can sort by your average search position, clicks and click-through rate to find potential improvements. "Links to your site" is also pretty useful.