You need to compile a Research and Analysis document which will include the scope and introduction of the new network(s) and will define what the needs and requirements of both merging companies are and identify where bottlenecks may exist now.
You need to establish how you will do the following:-
Network topology and connectivity, speed
Back-up and disaster recovery
Security - Firewalls, VPN's, Extranets, UTM devices, Proxy servers, Malware prevention etc.
User and administrator rights
Maintenance, monitoring and updates
Upgradeability, future proofing.
Costing, suppliers and whether to buy or lease.
Software/licensing tracking
Until you do all this you are flying blind and only guessing what is required to build your network which cannot be done without expert knowledge or in the ad hoc way that you are implying in your question. You may require expert help on this one.
You need to establish security zones. The most secure being your LAN/intranet and followed by the DMZ or Extranet where you can connect both the inner and outer Worlds and with mobile infectable devices and finally the least secure, internet.
I would recommend establishing an internal secure LAN in the parent company in another city and connect it to your office via a VPN tunnel which is highly secure.