For that kind of use I think you will notice the difference between the phones a lot more than the source. Main choice you have to make is do you want to be able to hear your surroundings, which can be important if by bike you mean bicycle, or shut them out which you would probably want more if bike means motorcycle. I don't do lo-fi so can't help you with buds but when it comes to quality IEMs the Shure SE535 is kind of a gold standard, the IEM equivalent of the Sennheiser HD650. The SE535 seals you off with something like 20 dB isolation. They will also eat up almost all your budget. Which is fine, because all you need with them is a little old iPod. They come with a pretty nice case that depending on which iPod you get the whole thing phones and iPod will fit.
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Well I just bought a Chord Mojo and Cardas EM5813 Model 1 Earbuds from Audio Advisor both were on deal and the salesman says its about as good as gets with either an Iphone or Android. Should be a good 6 to 8 hours without a recharge. We shall see. They both have great reviews so Im kinda excited to be able to blank out humanity when Im shopping or on a bike ride. Haha. And Thank you especially Millercarbon you have always have detailed, knowledgeable, and thoughtful responses. Happy early Fourth of July to ya!
I had enjoyed using SoundMAGIC E11C while out of town for 3 weeks on May using Galaxy Note 4.
After being back to my home, I had it connected it to my Chord Dave and Mscaler.
Wow it sounds wonderful with neutral balance, treble and dynamics.
This one is a best buy earbud.
For $70 the Fiio BRT3 is amazing. Bluetooth only, but it is one of the best sounding portable DAC/Amps I've ever heard.
With the right source it can have an amazing mid/treble liquidity I've only heard before on the Pono.