This configuration is known as split tunnelling - your VPN client only sends traffic destined for the office network over the VPN, and saves money in bandwidth utilization (and speeds up your web browsing) by not routing traffic destined for the internet over the VPN and out through the office web proxy.
A SSH tunnel is a point-to-point connection, not a method of internet access, and a SOCKS proxy is a server that acts as a go-between between your PC and the internet and is generally not available to home users.
Secure and private browsing is a pretty deep and broad field of knowledge - but it will require endpoint security (security software on your PC or a firewall you set up between your home network and the internet.) Home and home-office users generally rely on security software installed on their PC - if you have a Windows PC, you will need, at a minimum, decent anti-malware ("AV") and firewall software.
For secure browsing, here is a good overview for beginners.