I'm going to write a few thoughts about recreating specific synth sounds that your hear in songs and want to recreate yourself. If you only have access to limited gear this can be great as a lot of tutorials you find on the internet will focus on synths you might not have; also it's really good to have one synth that you know well enough to recreate all your favourite sounds on.
"Give a man a fish and you feed him for a day; teach a man to fish and you feed him for a lifetime." (Chinese Proverb)
First things first, research the artist and try to identify which synths they use. Look at studio pictures and see if you can identity synths as Moogs, Rolands or Korgs. Read some interviews and see if they namedrop their favourite bits of gear. Do they use analog hardware or do they work on laptop-based setup that would more point to the use of soft synths? Look at live videos and see what equipment they use to recreate their sounds. Do they use effect pedals? Finding answers to these question can really help take some of the guesswork out of recreating their sounds.
2. Listen to the Whole Song
Oftentimes a synthesizer sound will change throughout a song and may reveal itself better in a certain part of the song. A great example is 'Everything In It's Right Place' by Radiohead, which is built around a synth part that sounds strongly like an electric piano, maybe a Wurlitzer or a Rhodes. However towards the end of the song the filter opens up and we can hear clearly that it's really a rich sounding polysynth that has it's filter right down at the beginning of the song. Doing some research you can read that they were using the Sequential Circuits Prophet 5 heavily around the Kid A era. This could very well be that synth used in 'Everything In It's Right Place'.
3. Brightness vs Darkness
Listen to the below examples and try to work out where the filters are set. Listen for a high pass filter, a low pass filter, and resonance in these filters. If you don't know what these sound like then experiment with the filters on your own synths. Listening to the bass notes can be a giveaway for HPFs. Here are the examples:
Do this whenever you're listening to a synth part with the aim of recreating it, as this is a really important part of a synth sound.
Next time you're jamming on a synth, be it hardware or software, try switching the oscillator waveforms between sawtooth, square, triangle and pulse and listen to the effect that this has on the timbre of the patch. Try listening to how each waveform sounds through a bright filter or a dark filter. Listen to the waveform through short and long envelopes. The more you can tell the difference between each waveform the more you'll be able to listen to a synth part on a record and immediately tell which waveform is being used.
Sometimes the sound is a really FX-heavy sound, and the actual synth settings have less to do with the sound then the effects. Learn to recognise chorus (such as the built in Roland Juno chorus), phasers and delay and these sounds will become easy to recreate. Often in these cases the specific synth used isn't particularly important but the type of effect is.
Recognising and recreating synth sounds is a skill just like playing guitar solos or keeping a drum beat in time. The more you practice creating synth sounds the more your ears will develop and the more effortless it will become. You'll gain the skill of being able to hear sounds in your head and know how to translate them to a synthesizer, and you'll also be able to recreate the same sound on multiple synths and recognise the nuances of individual synthesizers.
I encourage every aspiring synthesist to work on developing this skill. Use the comments below to share what you're working on and any sounds you're struggling with and I'll gladly offer some help.