Experience is critical because making mistakes ends up costing a lot of money. Bad architecture decisions can have huge impact upon throughputs, etc.
Based on my experience, the best way would be to get a job at a medium size systems integration firm. Your ability to write scripts etc might get your foot in the door. Then you would have to learn fast from the experienced infrastructure engineers.
Of course, you will also need to read a lot and learn a lot about LAN, WAN, SAN, VPN, DNS, firewalls, virtualisation, cloud, etc, etc.