Your question is far too broad to cover in the GIS.SE Q&A format. It sounds like you're at square one with little to no previous knowledge or experience. This is fine, everyone has to start somewhere. But if you really want to learn the proper methodolgy, you're going to need to do some research into all the various steps, even just to get an overview of the process - let alone the finer points.
Step 1: Surveying
Before you go out to collect data, you need to understand a few things about designing sample collection. There are some great tutorials out there about how to do topograpic surveys of a piece of ground. The old way to do it is called differential leveling, but even those tutorials have instructions on how to lay out survey points. You would just be GPSing those points rather than using a transit on them.
For topography, setting up a reasonably regular grid of points, and then collecting a few more at outliers (like a low spot around a drain inlet, the high point of a hill, or along the bottom of a swale) should get you fairly decent data to work with. Just walking around randomly to collect points isn't going to produce a very good surface.
As mentioned in my comment above, you need to be careful with the unit you have and how you use it. If you have a true survey instrument, it has a pole and you should use it - only collect points with the pole on the ground. If you don't have one, just attach the gps to a walking stick or something so you're always reading at the same height. Of course, if using a consumer grade unit (or method, ie no base station), the unit accuracy isn't going to be great in the vertical to begin with. A less accurate consumer unit might be better off with a larger area that has greater variation than say a small field or quad that might have a ten foot or less elevation range.
For that matter, GPS readings can vary quite a bit - if you really want some consistent data, you need to record several readings at the same point, and average them together (some units do this as an optional, built-in function of collecting a point).
Step 2: Surface Generation
Once you have the points, you then create the surface from them either by raster interpolation or creating a TIN. There are several ways and formats of doing this in ArcGIS. For education, I'd recommend trying them all because they don't take very long. Fiddle around with each method. See why spline may or may not be better than kriging or IDW. Play with the input settings such as search radius, cell size, and weighting. Generate contour lines from your points, not just a surface. Make both a raster DEM and a TIN. Note the differences in format, what they look like, and how they behave. Which one processes/views faster. There are different tools that produce the same type of output in different ways. Try them all, compare the results.
I created the below maps/observations from a set of points in less than an hour, and that was including running each method a few times with varying settings to look at their influence on the output.
If you get stuck on a particular step, come back and ask a specific question. But have fun - I do this sort of thing both for work and entertainment.