Mililani is a large network of mostly single track that crosses PRIVATE land. It is not my recommendation to go there without a local who knows their way around, very easy to get lost in there. Laie is a public trail. I like fast downhills, and that is what you get at Laie. It's a long grinding climb to the "trees" at the halfway point of the trail. The trail continues on to a waterfall, but most bikers stop at the "trees" and head down from there. Straight up and straight down, that's Laie. You can ride that one on your own, it's well marked and easy to follow.
Now as to rent or bring your bike. I have traved many times from Hawaii to the mainland and have brought my bike each time. I really beleive that the riding and my enjoyment of the riding were completely enhanced by bringing my own bike. Yes it can be expensive, but the familiarity of the bike helps make you much sharper when riding unfamiliar trails.
There are lots of other trails to ride on Oahu too. Yes January is a "wet" month, but if you ride on the Leeward side (Makakilo, Yokohama) you can avoid the mud bath. We are extra sensative about riding when it's very wet, so please kokua and road ride on super wet days.
Check out DHHawaii.com, RideHawaii.com , hawaiitrails.org for more info.