Can someone point me to some good thorough tutorial how this can be achieved? Or simply post few steps if it's not too cumbersome?

Here's my setup:
Machine 1:
eth0 (connected internal network and routed to Internet through router)
eth1 (use it as a bridge)

Machine 2:
eth0 (connected to eth1 on Machine 1)

Correct me if I'm wrong, but I understand no ip address should be assigned to eth1 on Machine 1, and on Machine 2, ip from internal network should be assigned.

So setup is pretty simple but it's my first time trying to make this... I see also there's different "gateway" mathod with iptables. First one seems cleaner to me, but other one is acceptable if it's easier to setup...

A Linux computer is not a good switch - too slow, consumes too much energy and generates too much heat, too consumes too much space, few ports. Simple 4 port switches cost 20$ new, and you can probably get used ones for practically nothing. The cost of the switch will be offset by the saved electricity soon.

The Linux bridge use case is connecting virtual machines to the outside world.

Having said that, you will have to create the bridge and add both eth0 and eth1 to it. This means that the physical interfaces will not have their own IP address anymore; the bridge will have the IP address.

Here is a description how to do that in Debian: https://wiki.debian.org/BridgeNetworkConnections
And in Ubuntu for connecting VMs: https://wiki.ubuntu.com/KvmWithBridge

That machine, which will be used as a bridge is used non-stop anyways, so in my case the difference is electric consumtion of additional pci network card vs. 4-port switch. Thanks for links, will check them and will get back here in case of problems

Debian tutorial worked flawlessly on Kubuntu 14.04! It's really simple... Thanks!

Great!