This is a very old question, but I had many of the same questions in figuring out how the Internet works. Like the other answers, the networking books give an overview of BGP and DNS but still left me confused. For example, a.root-servers.net through m.root-servers.net are given as the root servers, but how does a DNS service know where to find those servers if they can't use DNS themselves.
The basics of IP, subnetting, DNS, etc. are assumed to be known by this answer. I am addressing "gaps" I, and probably the questioner, have on how the Internet works. By no means am I an expert, but this is my understanding of the gaps.
The first thing to note is that when the Internet started out as ARPANET, everybody knew everybody and routing tables for IP addresses were handcoded. I assume the assignment process for IP's was done over the phone. As the Internet became too big, BGP was used by multiple networks (AS's) to advertise they had public IP's or could get to a public IP through their AS to another AS. The trust was there that an AS wouldn't advertise an IP they didn't have.
Today, there's not as much trust. Instead, ISP's can download and authenticate the IP allocations to each AS from IANA and the regional authorities. These downloads are now authenticated through public key cryptography. So when IANA "assigns an IP address," they are changing their record (or really the regional authority changes their record). All other AS's can download and authenticate their records.
These records are important because ISP's can't take the word of other ISP's that they have the IP addresses. The ISP's can compare the BGP advertisement with the authenticated IP records. If any BGP advertisement shows the last AS as an AS other than what's in IANA's and RIR's authenticated record, the BGP advertisement does not change their own routing.
More commonly, a rogue ISP or AS can advertise they have a route through their AS they don't have. AS1 has an IP registered and AS5 currently uses AS5 -> AS4 -> AS3 -> AS1 -> IP. AS2 advertises to AS5 a route of AS5 -> AS2 -> AS1 -> IP. Except AS2 doesn't actually have a connection with AS1. It can just lose the packets, maybe to frustrate AS1's hosting customers. Or AS2 could be a small company network with a multihomed arrangement with AS5 and AS1. Their router is misconfigured and advertises a path through a small company network. Nearly all ISP's throw away such advertisements of their BGP customers and only pass on terminating BGP advertisements.
More likely, you have the case of Pakistan trying to shut off Youtube in Pakistan through such IP hijacking, and shutting off Youtube outside of Pakistan too since AS's outside of Pakistan assumed their BGP advertisements were correct.
In the end, there isn't a perfect defense against such IP hijacking. In most countries like the US, such abuse of BGP can be punished as breach of contract and other ISP's will shut off peering connections with that AS if they have to. An ISP could also disregard the whole IANA and RIR apparatus and redirect the IP addresses to their own servers. That won't work for any https sites though, assuming the ISP doesn't have the private keys for any CA's. There is very little to gain from it economically. It only happens with authoritarian governments, such as Egypt recently shutting off all BGP advertisements to their ISP's from outside the country.
DNS is somewhat simpler once the IP tables are correct. The root servers are all hardcorded IP addresses in the DNS server code. a.root-servers.net is 18.104.22.168 and the IP address is anycast within one AS. In the case of a.root-servers.net, the AS is Verisign and there are five different sites. In the US, the two sites are New York and LA. Anycasting is like if you had an address of 123 Main Street and you said "It doesn't matter what town you are in, go to 123 Main Street and you'll find one of my businesses." Both 123 Main Street in NY and LA will give the same answer for all top-level domains. The AS, in this case Verisign, figures out internally which server has the fewest hops through OSPF, internal BGP, and other routing protocols. So a router in Denver may go to LA while a router in Chicago goes to New York. The same routing process can be used for Anycast hosts because the hosts don't offer to route traffic.
One of the root servers gives which IP address for the com top-level domain. Then that domain gives the domain for yoursite.com. The registrars really have a contract with whoever runs the top-level domain. So if the top-level domain currently doesn't have a record for yoursite.com, it has access to add a record with their who-is server. Then, with the access the registrar gave you to yoursite.com's DNS records, you change the records in their DNS server to go to your IP address.
Because DNS all depends on multiple IP addresses going to the right place, you have the same issue as before with AS's authenticating the IP registry and then the BGP assignments. That is the key piece for an http website. Https has the added protection of certificates. So, an ISP can't reroute requests for their own root servers and top-level domain servers to give their own IP for, say, citibank.com. If they did, the IP address given to the user will be a different IP address, but their server won't have Citibank's private key.