How can I log in and get access to my data from any computer? Is it safe?
If they are your computers, just install the plugin on all browsers you plan to use with LastPass: http://lastpass.com/download.php.
If you're on a friend's computer or traveling, you can access your sites from https://lastpass.com/ at any time. Though the plugin is required for LastPass to autofill and autologin to sites, you can still copy and paste your credentials from the Vault.
Whenever you setup, edit or add sites and other data, your sensitive information is encrypted locally on your computer before the data is sent to LastPass.