Defining what you mean by "malicious", what you mean by "investigate", and who is doing the investigating are all important.
If you need to determine if the site is hosting malicious code, then you need to run a malware scan on the URL. There are many online sites that can do this for you (as well as the AV on your machine). VirusTotal is a big one and probably the most famous. It will give you an opinion that the average user can understand.
There are also sandboxes that you can set up that inspect what the code on the linked site does so that you can do an in-depth analysis on the site yourself. This is not for the average user.
Phishing defense sites, like PhishTank, will help you look up to see if the site has phishing markers or if the link has been reported as a phishing site by the community.
There are also various reputation ranking sites and URL blacklist sites that allow you to look up how "dodgy" a site is based on its past behaviour.
For shortened URLs, you can use a URL de-shortener to find out where the link actually goes to then feed into the above.