What is the best way to read, represent and render map data?
SharpMap is an open-source .NET 2.0 mapping engine for WinForms and ASP.NET. This may provide all the functionality that you need. It deals with most common GIS vector and raster data formats including ESRI shapefiles.
First, I recommend that you use the 2008 TIGER files.
Second, as others point out there are a lot of projects out there now that already read in, interpret, convert, and use the data. Building your own parser for this data is almost trivial, though, so there's no reason to go through another project's code and try to extract what you need unless you plan on using their project as a whole.
If you want to start from the lower level
Parsing
Building your own TIGER parser (reasonably easy - just a DB of line segments), and building a simple render on top of that (lines, polygons, letters/names) is also going to be fairly easy. You'll want to look at various map projection types for the render phase. The most frequently used (and therefore most familiar to users) is the Mercator projection - it's fairly simple and fast. You might want to play with supporting other projections.
This will provide a bit of 'fun' in terms of seeing how to project a map, and how to reverse that projection (say a user clicks on the map, you want to see the lat/lon they clicked - requires reversing the current projection equation).
Rendering
When I developed my renderer I decided to base my window on a fixed size (embedded device), and a fixed magnification. This meant that I could center the map at a lat/lon, and with the center pixel=center lat/lon at a given magnification, and given the mercator projection I could calculate which pixel represented each lat/lon, and vice-versa.
Some programs instead allow the window to vary, and instead of using magnification and a fixed point, they use two fixed points (often the upper left and lower right corners of a rectangle defining the window). In this case it becomes trivial to determine the pixel to lat/lon transfer - it's just a few interpolation calculations. Rotating and scaling make this transfer function a little more complex, but shouldn't be considerably so - it's still a rectangular window with interpolation, but the window corners don't need to be in any particular orientation with respect to north. This adds a few corner cases (you can turn the map inside out and view it as if from inside the earth, for instance) but these aren't onerous, and can be dealt with as you work on it.
Once you've got the lat/lon to pixel transfer done, rendering lines and polygons is fairly simple except for normal graphics issues (such as edges of lines or polygons overlapping inappropriately, anti-aliasing, etc). But rendering a basic ugly map such as it done by many open source renderers is fairly straightforward.
You'll also be able to play with distance and great circle calculations - for instance a nice rule of thumb is that every degree of lat or lon at the equator is approximately 111.1KM - but one changes as you get closer to either pole, while the other continues to remain at 111.1kM.
Storage and Structures
How you store and refer to the data, however, depends greatly on what you plan on doing with it. A lot of difficult problems arise if you want to use the same database structure for demographics vs routing - a given data base structure and indexing will be fast for one, and slow for the other.
Using zipcodes and loading only the nearby zipcodes works for small map rendering projects, but if you need a route across the country you need a different structure. Some implementations have 'overlay' databases which only contain major roads and snaps routes to the overlay (or through multiple overlays - local, metro, county, state, country). This results in fast, but sometimes inefficient routing.
Tiling
Tiling your map is actually not easy. At lower magnifications you can render a whole map and cut it up. At higher magnifications you can't render the whole thing at once (due to memory/space constraints), so you have to slice it up.
Cutting lines at boundaries of tiles so you can render individual tiles results in less than perfect results - often what is done is lines are rendered beyond the tile boundary (or, at least the data of the line end is kept, though rendering stops once it finds it's fallen off the edge) - this reduces error that occurs with lines looking like they don't quite match as they travel across tiles.
You'll see what I'm talking about as you work on this problem.
It isn't trivial to find the data that goes into a given tile as well - a line may have both ends outside a given tile, but travel across the tile. You'll need to consult graphics books about this (Michael Abrash's book is the seminal reference, freely available now at the preceding link). While it talks mostly about gaming, the windowing, clipping, polygon edges, collision, etc all apply here.
However, you might want to play at a higher level.
Once you have the above done (either by adapting an existing project, or doing the above yourself) you may want to play with other scenarios and algorithms.
Reverse geocoding is reasonably easy. Input lat/lon (or click on map) and get the nearest address. This teaches you how to interpret addresses along line segments in TIGER data.
Basic geocoding is a hard problem. Writing an address parser is a useful and interesting project, and then converting that into lat/lon using the TIGER data is non-trivial, but a lot of fun. Start out simple and small by requiring exact name and format matching, and then start to look into 'like' matching and phonetic matching. There's a lot of research in this area - look at search engine projects for some help here.
Finding the shortest path between two points is a non-trivial problem. There are many, many algorithms for doing that, most of which are patented. I recommend that if you try this go with an easy algorithm of your own design, and then do some research and compare your design to the state of the art. It's a lot of fun if you're into graph theory.
Following a path and pre-emptively giving instructions is not as easy as it looks on first blush. Given a set of instructions with an associated array of lat/lon pairs, 'follow' the route using external input (GPS, or simulated GPS) and develop an algorithm that gives the user instructions as they approach each real intersection. Notice that there are more lat/lon pairs than instructions due to curving roads, etc, and you'll need to detect direction of travel and so forth. Lots of corner cases you won't see until you try to implement it.
Point of interest search. This one is interesting - you need to find the current location, and all the points of interest (not part of TIGER, make your own or get another source) within a certain distance (as the crow flies, or harder - driving distance) of the origin. This one is interesting in that you have to convert the POI database into a format that is easy to search in this circumstance. You can't take the time to go through millions of entries, do the distance calculation (sqrt(x^2 + y^2)), and return the results. You need to have some method or algorithm to cut the amount of data down first.
Traveling salesman. Routing with multiple destinations. Just a harder version of regular routing.
You can find a number of links to many projects and sources of information on this subject here.
Good luck, and please publish whatever you do, no matter how rudimentary or ugly, so others can benefit!
-Adam