Welcome aboard Jonathan and congrats on making a sound financial decision for you and your family. And God Bless the family that gave you a gem of a vehicle.
We're on our 5th MDX and have owned 9 Acuras in all. Our first MDX was a 2006 Touring which is almost exactly what you have (2005 Touring). We sold our 2006 at 206,000 miles but it was still going strong. Like many on this board, I did most of the mechanical work to the 2006 and can offer some advice for sure.
First of all, IMO, the 2005-2006 and 2018-2020 models are the best MDX's to own. Almost all MDX'es are bulletproof, but these models especially had very little issues. Your 2005 should go for many years; some on here have gone 400K+ miles.
Besides the obvious
oil change, new
battery,
brake rotor and
pad inspection, and
tire inspection, as Obi1 suggested above, the
timing belt and
water pump are the first things to have checked out. Timing belt swaps are recommended every 105K miles. Because the timing belt is right next to the water pump, it's usually recommended to have the water pump done at the same time.
Another inspection point would be the
spark plugs. Especially if the vehicle was sitting for a few years, new plugs might be a good idea. Have your mechanic check the
spark plug wires as well. The
alternator would be a another check-point as well (again, considering the lag time sitting in the driveway).
CV joints,
engine mounts (although you can live with cracked engine mounts), and a steering check to listen for whining would be a good idea. Whining while steering can be a sign of the o-ring in the
steering pump needing to be replaced (very common), but it could also be a sign that your
rack and pinion is going out. With that mileage, these are considerations.
I could go on further, but given the age of the vehicle and the small market value, at this point, I recommend doing the basic maintenance items and not "fishing" for problems. Instead I would just address issues if they arise.
As a mobile detailer (during my college years), I can tell you a thorough exterior buff and clean and an interior cleanse would do wonders. I'd immediately swap out all of those non-OEM mats and buy
all-weather Acura or WeatherTech (type) mats that conform to the vehicle's interior. That rear MDX cargo mat just needs a thorough cleaning - it can be salvaged. Those interior stains can be greatly reduced by a skilled detailer.
The
leather seats in the rear look good - the front seats have the usual wear and tear. The good news is that you don't have to replace the whole seat - you can buy just the bottoms.
Here's an example of both front seats OEM style replacement covers. It's about $100 more than if you go with leather-like seat covers, but honestly, the OEM replacement of the bottom leather is a way better solution. I bought the Iggee leather covers (for the whole seat) and I wish I had gone with the OEM bottom covers.
I think that's a good start to get you on your way...congrats and good luck!