I would suggest an approach that heavily focuses on simple, descriptive statistics and visualization (and largely ignores the flow part of the data at the onset).
First examine the prevalence of traffic on all of your streets via map visualizations (proportional pin maps, kernel density, choropleth streets etc.) and frequency tables/histograms. I would suspect just examining this would allow you to have pretty good guesses as to the nature of the flows on the network. I would also say that you won't lose much by visualizing points as oppossed to the entire streets, although if convienant you might as well generate both types of maps. At this stage I would focus on the areas that have a high prevalence, and any interesting contrasts that appear (i.e. streets of little to no traffic next to streets with high traffic).
Next steps would depend on what interesting questions you either currently have or develop. Dyadic connections between the streets would be fairly simple to examine, but any larger connections are likely difficult to uncover without more complex strategies and specific hypothesis. Example questions could be "Do bikers avoid steep hills?", "Do bikers avoid stop light intersections?", or "Do bikers tend to use roads with a bike lane more frequently?". Such hypothesis may seem plausible after examining the prevalence of bike trips on streets, especially if you have local knowledge of the area and observe contrasts supporting those hypotheses.
If you have covariates that you suspect may have some relationship with bike riding at the onset of the analysis you can repeat all the same suggestions I gave above except condition the maps and other statistics on those covariates and attempt to identify any interesting changes in patterns. An example might be "Do people take different bike routes on the weekend than during the work week?". For this you might develop two maps, one with bike trips taken from Monday to Friday compared to bike trips on Saturday and Sunday. If the covariate is continuous, scatterplots of the covariate against the X and Y coordinates of bike trips might identify any obvious changes in spatial patterns (that is a scatterplot of the covariate against all bike trips as oppossed to the number of trips on a certain street segment).
For temporal visualization I would suggest exporting the data to kml with time stamps if possible and using Google Earth. I really enjoy the time clock in GE (much more so than any add-in for ArcMap that I have found), and it has the added benifit of the aerial photography and the ability to search nearby locations if you decipher any interesting patterns. I have not tried this with line segments, but it works quite fast for point data and produces a really cool display (it did not work very fast for polygon data the last time I tried). Otherwise producing small multiple maps or sequential images is another option (although not as fun as the animation in GE).
When you develop more specific questions, potentially interesting ways to analyze the data may be with simulated agent based models or discrete choice models. But these will only be necessary if you have more complex hypothesis, and they would only have a limited value for exploration IMO. Such data can be really complicated, and going gradually from simple to more complicated I believe is the best approach to not become overwhelmed.