First, there's lots of examples to start from. As Picasso once said, "Good artists copy; great artists steal."
Let the theft begin, with a search for "ekg circuit" or "ecg circuit". Most examples you will find are the 3-lead type with a driven 'leg' signal. Others are of the basic instrumentation op-amp type with 2 terminals. Sift through these and see what makes sense for you. They will usually include some kind of bandpass filter as well.
Second, before you commit to a design, build a simulation of it, at least for the analog part. There are good free Spice simulators like LTSpice and MicroCap. You may even be able to get the job done with Falstad. Falstad in particular will allow easier observation of your circuit. It's a good starting point for developing a proof-of-concept with practically no learning curve.
The 'real' Spice simulators like LTSpice, PSpice and others will provide more accurate models of your components, especially if you choose Analog Devices parts and are using LTSpice (that's part of ADI's marketing, and you benefit from their well-maintained simulator.) Expect some learning curve.
As for your specific questions about filter design, you can also work that out in simulation. Again, copy if you can, but if you're determined to develop your own, know that you will need to first understand your requirements, then choose a filter basic type (bessel, butterworth, chebyshev, elliptical) that makes the appropriate tradeoffs in passband flatness vs. cutoff slope. Fortunately, there are online tools for that too. Also, read up on filters here.
I'll mention that Falstad also has an analog filter design applet separate from the analog simulator that makes it easy to try out filters. You can cut and paste between the filter app and the circuit sim.
A tip: For an ECG/EKG, your signal rate is low enough that it would be worth investigating doing some of your filtering in the digital domain on your microcontroller. This could simplify your front end design down to just a gently-sloped anti-aliasing filter prior to the ADC input. I think a Cortex-M4 should have enough throughput. Read more here. And, an NXP Appnote doing just that.
(How would one do this "professionally"? Probably using Matlab and Simulink. That's a whole 'nother ball of wax.)
Once you're satisfied with your simulation, then it makes sense to lash up a prototype and test it to confirm what you see in the simulation. Build this with care, taking into account your concerns about noise. This is also where you get to play with how to make good skin contact. A conductive gel helps - you can even make your own from an aloe plant and table salt.
Also, for your first go don't use line power, use a battery. Even an 'isolated' supply will have some AC leakage that will couple into your measurement, and there is some safety concern too. (Note that medical supplies are designed with lower leakage, for both signal and safety reasons.)
Once you have all that settled, you'd capture your design in a schematic, create a layout and make your board. KiCAD is a free EDA suite that can help do that.