Just some thoughts.
Go to the company that OWNS the data center. This is a preference thing for me, but if you go to a company that is leasing space in the data center and then re-selling, you have 2 levels to go though if there is an issue and likely you can't even talk to the real owner of the data center. I prefer to deal with the true owners. There are plenty of companies out there that work by leasing space and do a good job but this is my preference.
Make sure the contract indicates you own your equipment. If the company goes out of business, you want to be sure you can get your equipment back, even then it might not happen. This relates to the top one, if the company leasing the space from the DC doesn't pay their bill, all the equipment might be seized including yours. This happened to me, lucky the main company knew this and contacted us (and the others I assume) and we were able to do a contact with them.
Consider less obvious locations. Cable companies for one. They are doing VoIP services now so they need huge bandwidth, but are not using it all. They need to have really high SLA for their cable customers and the VoIP so they usually have a really good setup. I got a good deal at one, paying about half of what I would have at a high end place. The only downtime they had is when about half the internet became flaky.
If you are going local (thats what I prefer) get a tour. See it and what they will give you. Sometimes you can get a better deal. With the cable place I actually was able to negioate.
If you can, put one server there (a non critial one) for a month or 2. Make sure everything is good. The place may put up a good show when you are there but then you find out their sevice sucks, or even worse is down a lot. This one I learned from a bad experience.
Since this equipment is out of your control physically make sure you can afford to lose it. It shouldn't happen, but it could. So have a disaster recovery plan in place so that if your server suddenly go offline and you can't get into the place to get them back (or fire, flood, etc) you have a way to bring up servers at your own office in the interm. VM is great for this.
Normally when changing the public IP of a website I like to run the site on both IPs for at least a couple of days as there are lots of people out there who have cached IPs in their DNS servers and they'll keep getting the old IP. Using this approach, assuming that when Dreamhost removes the website from DNS they also shutdown the old site on the old server people using the cached (now wrong) IP would get an error message instead of the site.
The worst I've seen was when moving one site for a client we had people hitting the old server for a month (just a couple people). We ended up having to call those clients to have them flush their DNS cache manually.
Best Answer
Yes, you can do that. It is called colocation. Essentially you provide the server and the colocation provider supplies everything else: power, cooling, security, and in some cases they provide bandwidth in some cases you can provide it yourself.
They will base the cost on how much physical space your server takes up, how much power it uses how much heat it generates and if you need bandwidth, how much bandwidth you will be using. Typically if they provide bandwidth they will charge you a fixed amount for a certain number of IP addresses and a set CIR (committed information rate). You can either pay for a fixed amount of bandwidth or you can pay based on usage. If you pay for fixed bandwidth then they will give you that amount and you will never be able to use more. It is quite common for you to buy burstable bandwidth though. In this scenario they will provide you with a port that can go up to 100 Mbps for example and bill you based on your average utilization. Typically this is done using the 95th percentile billing model (Google can explain it better than I can). So they may charge you $50/mo per megabit of 95th percentile bandwidth so if you average 10 Mbps then you would pay $500 for that month.
Having said that, if you only have 1 server then it would probably be drastically simpler and easier to either rent a dedicated server or use a virtual server (VPS). Companies such as Rackspace and Amazon Web Services provide virtual servers. In that model you would pay for the virtual server based on how much CPU, RAM and disk you need. You also pay for bandwidth but in this case you pay based upon how much data you transfer, not your average utilization. For example, AWS charges about 10-12 cents per gigabyte of data your server sends to the Internet.
There are other advantages to using a virtual server. You no longer have to worry about hardware failures since your server is a virtual machine if the host it is running on has a hardware problem it can easily be moved to another host. Additionally it is easy to upgrade the virtual machine to have more or less CPU/RAM/Disk based upon your usage.