Here's our list of important contacts:
HVAC - Units only break down when you need them most. We agree with @Jonathan G. about keeping filters changed regularly. Don't assume your tenant will do that for you even if you leave a supply of filters onsite.
Plumber - We do almost all plumbing work ourselves. But we often buy houses that have been sitting empty for some time and have plumbing lines that have become blocked with tree roots and debris. Plumbers have equipment that is much more heavy-duty than what we own ourselves and are experts in rooting out plumbing lines and ensuring drains flow freely.
Electrician - While we do quite a bit of electrical work ourselves, it is important to have an electrician available for more complex work, such as upgrading an electrical box or replacing wiring.
Carpet Sales and Installation - Every property we have ever owned has received new carpet at some point during our ownership. Find a carpet dealer and installer you trust. We use an agent who provides in-home service. It is super convenient and provides significant savings over traditional carpet stores.
Sheet Rock Crew - Most DIY'ers can do sheet rock work pretty easily. Patching holes or retexturing a room should be well within your own capabilities. But often we need several rooms or even an entire house done. A good sheet rock crew can save you weeks of time and complete your project much more efficiently.
Tree Trimmer - Minor pruning is a piece of cake. But for big jobs, we almost always use a professional tree trimmer. In our experience, tree trimming costs vary wildly among contractors. Get several bids before settling into a relationship.