I write for this site:
which sometimes, when my connection is a bit slower, needs over 30 seconds to load. My question is, which online tool can give me the best and detailed instructions about what to change.
I've found these two:
but is this worth to try?
There are only two ways to make a web page load faster - make the files smaller and reduce the number of files.
One thing that's often recommended is to gzip your files, which is also recommended in the first link you posted. You can do that easily enough by adding some code to your .htaccess file. You could grab the code used in the HTML5 boilerplate as an example.
The weight of the homepage is just 223KB running on the default 2012 wordpress theme. It's not a content, multi-file, or filesize issue.
I'm might suspect hosting issues (overloaded shared server, maybe?). Who are you using as a hosting provider?
I'm not sure where the site is hosted. Anyway, the gmetrix.com tool mentions two things on a server level: "leverage browser caching" and "enable gzip compression." Does "on the server level" means that a hosting provider has to solve this or the site owner can do something?
No, that normally means you do it yourself. I mentioned how to do the gzip bit above. You access the .htaccess that's normally in the root folder of your site. (If there isn't one, you can create it.)
We did the gzip thing. The site now fully loads in 3 seconds...when it loads normally. It still needs like 8 sec on another bit slower connection, while one other "control" site (webmd.com) always loads with about the same speed.
I registered the site to alertra.com and got 2-3 e-mail warnings per day: "Site down." The site actually needed 0.5 to few minutes to load each time during those episodes.