On timing: BOOK EARLY. I'm not sure at what stages prices change, but when they do, they often go up drastically. In my last move, and the one I'll be completing soon, I nearly had to pay an extra $300 because I waited until around a month before my moving date to pull the trigger on the reservation. I was only saved by the fact that my moves were flexible and I could just push it back a week. Waiting seems even more stupid when you factor in that many companies don't charge you for your reservation until you show up to pick up the vehicle.
On choice of moving company: I really recommend Penske. Everyone knows UHaul, but for a few reasons they're inferior to Penske IMO. FWIW, the costs of these two options have been very comparable for me in the past. If there was a huge disparity with UHaul being better I might've made a different decision, but I'd pay at least a bit extra for Penske at this point given my experiences. Why does this matter? When you book with UHaul, you aren't guaranteed a truck. I've read plenty of reviews and even talked with family members who showed up to get their truck and they simply didn't have it, so were forced to either take a smaller/bigger one or come back another day. I know I'd be pissed considering how much most people plan out their moving day, so this seems really unacceptable when you can go with Penske, which does guarantee your vehicle. Secondly, the quality of vehicles with Penske is noticeably different. From what I've read, Penske rotates their trucks out of the rotation fairly regularly, while UHaul basically runs them into the ground until they're dead. I've used both companies and can say that my experience confirms this. I can't say too much on other companies, as many others I've looked up don't do one way rentals (only local) and that's what I've needed. If you have other DIY options, just be sure to research these policies.
There are certainly other options, but I figured I'd throw out some advice based on my experience/research. If anyone has experience with other companies or moving alternatives, feel free to chime in and add. Good luck.
