1 Audit/check/Inventory/review:

    General documentation: do not impove until you understand the environment (do backup before change)
    Crucial services inventory: Identify mission critical services and where they are hosted; identify who is responsible for them if its not you
    [DR] Disaster Recovery Plan: Backups are running properly? Backup rotation? Last DR test? Automated? In case of my absence?
    [BC] Business Continuity Plan
    [BIA] Business Impact Analysis
    Network topology: configuration (backup?), passwords, routers, gateways, subnets, vlans, static addresses, dhcp, labeled cables
    Power supply/UPS
    ISP: contact, agreements, SLA, contracts
    Support for environment's components: contact, agreements, consultants, SLA, contracts; renew/remedy any issues regarding lack of support, get replacement parts in a timely fashion, maintenance contract situation
    VPN / Remote Access
    Firewall policies: understand what's being permitted/blocked
    AV: existing on systems (servers, desktops, mobiles), activated, updated, custom exclusions
    Password repository: existing? Up to date?
    Admin accounts: running services
    Encryption certificates expiration date
    Windows Updates: policies, working?
    Applications updates: policies? automated?
    Software Inventory: licences (with charges), warranty, legal
    Hardwar Inventory: warranty, replacements parts, end of life cycle situation
    Scheduler jobs on servers
    GPOs review
    Scripts review
    Observe network/systems: to know what is "normal" behavior; known problems; check logs
    Study last audits reports.
    Process reviews for incidents, problems management, service requests, escalation [ITIL]
    [Optional]: Phone systems - VOIP;Skype for Business;other communication solutions/channels

2 Prepare/make

    Crisis kit: fire safe place, vendor contacts, emergency numbers, screwdriver, towel, deodorant, phone charger, headache medication, cable tester, takeout menu
    Meetings: with heads of departmentswhat their team does, what they use, what their major issues are
    Make a "Small wins": list that you can fix that will give you a bit of face to work with - this will contribute to people trusting that you're a professional there to provide a service.

3 Change

    Budget: now and in the future; limit extra useless PCs/laptops
    Categorize tickets: for future analizing
    Monitoring software: Icinga (or other software); iLo/iDrac sending mails; enable smart monitoring on disks, UPSes
    Clean up lazy permissions
    IDS/IPS (Intrusion Detection System/Intrusion Prevention System) if no existed

 certs expiring
Domain Names expiring
aintenance contract situation . Hardware end of life cycle situation. Study last audits reports.