Get all points of a street.

To highlight a street, you can use Overpass API.

The link

http://overpass-api.de/api/convert?data=(way[name~"Im Tannenbusch"](50.6,7.0,50.8,7.3);>;);out skel;target=openlayers

for example, highlights the street "Im Tannenbusch" within the rough bounding box north 50.6, east 7.0, north 50.8, east 7.3.

The link also shows that the question about the first and last point of a street is not so simple: there are three ends to that street. Other streets may have much more endings or even none at all if the street is circular. So a search for the first and the last point cannot be more than heuristics.

In general, you can get the raw data for a street with a link similar to the above

http://overpass-api.de/api/interpreter?data=(way[name~"Im Tannenbusch"](50.6,7.0,50.8,7.3);>;);out;

This raw data consists of the ways that comprise the street within the "way" tags. The "nd" tags correspond to the referred nodes (points). In this raw data you can look which of the first or last lines starting with "<nd" within a way-tag appears only once. These ids are the nodes at all the possibly multiple ends of a street.

Any basic frameworks about routing with OSM data can be found in the OSM wiki about Routing

But you have to define first how you will find the start and end node of a street, because one street can consist of several ways.

So tell us by adding a COMMENT here how you will define this.