EDIT: tl; dr version: Here's someone who built this on proto board. Have a look: http://egomachines.com/animoid/ann-hardware/build-full-adder.html

Now, more discussion: 
Gates have four kinds of connections:

 * Input
 * Output
 * Power
 * Ground

When you make a logic circuit using ICs, the first thing you need is to make sure they have their +5 and GND connections tied.

Once you've done that, connect your inputs and your outputs to the gate inputs and outputs.

Finally, if you have a gate output feeding another gate input, it's perfectly fine to tie them output-to-input. And, a single gate output can drive multiple gate inputs just by tying them directly.

Below is a diagram showing a full-adder connected from individual gates:
[![full adder from gates][1]][1]

Source http://www.theorycircuit.com/full-adder-circuit-diagram/

This shows the common power connections, and how the gate inputs and outputs tie together. The 7486, 7408 and 7432 are quad XOR, AND and OR gates, respectively. Below are detailed pinout diagrams for each:

[![Gate Package Pinouts][2]][2]

From here: http://www.theorycircuit.com/full-adder-circuit-diagram/

See if you can relate the connection to this diagram of a full adder from the link you posted:

[![Full Adder Diagram][3]][3]

From your link here: https://www.electronics-tutorials.ws/combination/comb_7.html

You'll see that it's using 2 XOR's (7486), 2 AND's (7408) and one OR (7432).

One final note: 74 series bipolar TTL logic (74xx, 74LSxx, 74Sxx, 74Fxx) defines an unconnected input as a logic '1'. This *does not* work for CMOS types (74Cxx, 74LVCxx), which requires all inputs, used or unused, to be tied to 0 (GND) or 1 (+5V / VCC.)


  [1]: https://i.sstatic.net/bbyD3.png
  [2]: https://i.sstatic.net/3QB0a.png
  [3]: https://i.sstatic.net/8T404.gif