The IANA assign IP address blocks to the regional internet registries (ARIN, RIPE, LACNIC, APNIC and AFRINIC) who in turn assign them to ISPs and other "autonomous" networks. Unfortunately most of them have run out of IPv4 address blocks for new regular allocations, so if you want IPv4 address blocks you have to buy them on the market and get the RIR to transfer them.
The smallest block of IPv4 addresses that is routed on the public internet is a /24 (256 addresses).
Once you have an IP block assigned, you will then need to interconnect with at least one (but preferably more than one) other ISP and exchange routes with them over BGP. There are two main types of interconnection (and some less common intermediate variants.
In a "peering" interconnection you exchange traffic from your network and it's customers with your peers network and it's customers. Peering is not transitive, if A peers with B and B peers with C, C does not get access to A. Peering is usually but not always "settlement free", that is neither network pays the other, though obviously someone has to pay for the infrastructure that carries the traffic between the two providers, this may be either a private link between the two providers or it may be an exchange point where multiple providers interconnect.
The other is a "transit" relationship, unlike peering this is asymetric, the transit customer pays the transit provider to deliver their traffic to and from the internet in general.
Most ISPs won't offer BGP peering or transit on "broadband" infrastructure, you will most likely have to pay for dedicated links to other providers or to internet exchange points. Sometimes it's more economical to rent space in locations where the providers you want to interconnect with already have a presence.
As a new network, particularly if you are on the hosting/content side, it is likely you will have to pay for transit services for the bulk of your traffic. Large ISPs will often refuse to peer with small ones, and even if they would peer with you, geography can become an issue. If you are on the access side things look a bit rosier as large content networks tend to have an open peering policy and a presence at a large number of locations and these can cover a substantial portion of your traffic, you will still probably have to pay transit for the rest though.
Finally most networks will require you to operate a 24/7 network operations center before they will peer with you. This means that running a network with peering can't be a single man operation.