The pump suction side is the center (end) and fluid ouput is the top. This video might be helpful:
https://www.youtube.com/watch?v=BaEHVpKc-1QThe gate valves can be closed in the event of replacing a pump, but should be open during operation. The plumbing is generally pretty logical: A bunch of hoses on the discharge sides which go to the jets (yours probably has diverter valves to turn some jets on and others off), and the water return comes through the filter. Look or large hoses on the return side. Problem is, it's hard to see all these things if the H/T has been sprayed with foam and the lines are all covered. It's helpful to look at hot tub plumbing that doesn't have any foam on it. Check out some of the hot tub manufacturer's videos on youtube that show the construction process of their hot tubs. There are quite a few good hot tub repair videos out there.
Like Tman said, the drain may be open since the neighbor's last actions were to empty it out. If not, you may need to put yours up on blocks or something to determine where any leaks are. There are gaskets at the pumps and heaters too. Also, if it's really old, you may consider disassembling the pumps and installing new seals. Cheaper to do ahead of time. There are parts on Craigslist from time to time if you live in an urban area.