in relation to "push the signal into the noise", he was talking about increasing the amplitude of the noise.
No, reducing the signal. There's generally not much you can do (directly) about noise)
Power is irrelevant. What is important is that you can discern the signal at the receiver.
The higher the SNR, the higher the data rate and the lower the error rate.
On top of that you can add error detection/correction protocols (which may involve retransmitting lost or corrupted data) which allow for a higher error rate (and thus lower SNR).
he meant that the lower the SNR the more distance it will cover. Am i on the right track?
Kinda. the SNR will be a limiting factor on your range. At some (presumably low) SNR you will not be able to achieve the speed you require.
As the distance between the transmitter and the receiver doubles, the signal will reduce to at most a quarter of its strength (by at least 6db). However, the noise will likely stay the same. So, your SNR will reduce by at least 6db for each doubling of distance. (that's an inverse square law)
If you have a transmitter/receiver pair that achieve a SNR of 60db at a distance of 10 metres, and require a SNR of 18db to maintain the data rate you desire, then your maximum range will be: 10 metres * 2^((60-18)/6) = 10 * 2^7m = 1.28 km
Your actual path losses can be greater. If you do not have line of sight, there will be losses due to absorption (say by trees) and reflection (it's not perfect). In addition t that, reflections can cause multipath distortion (especially at higher frequencies) and attenuation of the signal by subtractive interference -- causes dead spots).
You can increase your SNR by concentrating your signal, or by increasing the effective aperture of the receiving antenna (both actually mean the same thing).
So, without changing power, a transmitter using an antenna with 6db of gain, and a receive antenna with 3db of gain will increase the SNR by 9db.
Unfortunately, the coax you connect to the antenna will have losses, and so will connectors, etc.
For longer distances, you need to (typically) have antenna gain (especially receive antenna gain -- and I'll explain why later) and reduce losses/noise.
Picking a "good" frequency can help as one with lower noise is going to be better. Clearly you're constrained by the legal technicalities here, but even so, you can monitor the available frequencies to pick the "best" one.
Reducing losses in coax (typically making runs short and using good coax) is a "free" way to get more range.
(Oh, I didn't mention polarisation either)
Increasing transmitter antenna gain is good, but remember that this (a) makes it directional, and (b) increases EIRP which may have legal limitations. It is EIRP that generally limits your transmitter antenna's gain unless you want an omnidirectional antenna.
EIRP means that if you're putting 100mW into an antenna with 10db gain, it's like putting 1W into an isotropic (theoretical and perfectly omnidirectional) antenna. It only looks that way from one direction perhaps, but this is typically legally limited. Where I am, it is limited to 4W at 2.4GHz, so a 21db gain antenna (which I have on my roof) is limited to a maximum radiated power of about 30mW
Once you have this theoretical knowledge, you need to apply it to real life.
I suggest you look at some RF modules, probably looking at those with sensitive receivers. I would start with the NRF24L01 modules. These are good for several reasons:
1) they're cheap
2) they have library support for many uCs
3) there are (more expensive) versions with antenna connections
4) there are even versions with amplifiers (both receive and transmit)
5) they operate on spectrum that can legally be used in most places
6) they have inbuilt error correction protocols. You don't need to do it yourself.
7) you have control of the data rate so you can extend the range by reducing it.
I tried it with your example below: 
