Filter design for frequencies in range 0-60Mhz
As you stated there are some parameters associated with filters such as type of filter (butterworth, elliptic, chebyshev I,chebyshev II, bessel,...), order of filter, etc. (Note that there are lots of considerations in filter design bases on the application)

Let's first talk about the order of filters:
Filters with higher orders let you have a filter with better quality. For example in lowpass filter if you want to have a sharper transition band you need to design a filter with high order filers. Order of a passive analog filter is determined by the number of elements you use in filter (resistor, inductor, capacitors). This was all I knew about the order of the filter.

Now let's talk about the most applicable kinds of filter types. The most important difference between filter types is the band they have ripples. Note that the priority of filters is due to the application and when we say one filter is better than the other we must state that in what aspects (order, ripples, poles, zeros, stability, group delay, phase response, minimum phase, ...) this is better than that one. Picture below compares 4 important types of filters designed with the same order. As you can see, Elliptic filter has ripples both in pass-band and stop-band while it has the sharpest transition, and butterwoth filter has no ripples but has least sharpness among filters of the same order.