You have a couple of options, some may be harder than others, but should still work
1. Get a cable modem with a built in router and a built in switch (when I say router, I mean a device that not only has more than one ethernet port, but can do the act of routing using only 1 IP from your ISP, this will take your IP from something like 145.3.163.4 (I made that IP up) to something like 192.168.1.100 up to 192.168.1.255 (the routed IPs may differ depending on router brand) and avoid getting a cable modem with a switch but no router, a switch is not a router, meaning it will not make LAN IPs, and you will have to pay your ISP for 2 IPs, this will cost more)
2. Get a cable modem with a built-in switch, Even though I told you to avoid these, it could be easier since your ISP will take care of the DHCP IP leasing system, replace your original cable modem with it and connect one computer to see if it works (only one computer will work at a time, when one computer is online, the other computer will not connect), if you get the internet, it means that your service is working with the new modem, then call you ISP and request another IP, you will have to pay for 2 IPs, but you don't have to fuss around with the DHCP setting on the router
3. You can do the same thing as above using your current modem, you can either get a router and pay for only 1 IP, or pay for a switch and pay for 2 IPs, I would say go with a router, you do have to manage your own settings, but it costs less, to do any of these, just connect an ethernet cable from the modem to the router or switch, if you use a switch, call your ISP for an extra IP, if you get a router, just plug it in and set it up (should come with directions and your router login IP or domain)
4. You can do what you have suggested and get two modems if you wish, but you will need to get service for 2 modems, if you connect the second modem without paying for a second service line, only one computer can get online at a time, or the second modem may not even get service, talk to your ISP about how to use 2 modems, but you will have to pay twice as much per month
5. Like somebody else mentioned, you can use a crossover cable (an ethernet cable, that on one end, the green pair and the orange pair of wires are switched), get a second network card, connect the cable modem to one computer, install the extra card in the same computer, the take the crossover cable from the other computer to the dual ethernet computer, then set up internet connection sharing, and set your IPs, or you can install routing software on one computer, and have the DHCP in the routing software assign the IP for the other computer (I don't know of any routing software off the top of my head, I have only done routing on Windows Server, just do a google search for routing software), connecting one computer to another is harder, but it does work if you know what you are doing