Client
I'd go with OpenLayers - this is an opensource JavaScript client library, similar to the Google Maps API, or Bing API. However OpenLayers will free you of any licensing worries, or changing APIs as the source is available and can be modified - this is not the case with the Google/Bing/Yahoo APIs. You can however use the data from these services as layers in OpenLayers and drop them if they suddenly become filled with adverts or have commercial restrictions.
Have a look at examples to see if there is one you can use to make your first test application - http://openlayers.org/dev/examples/
GeoExtJS is a collection of extra tools and widgets that work with OpenLayers, built on top of another JavaScript framework ExtJS - also open source. ExtJS is mainly used for building user interfaces - data entry forms, grids etc. using JavaScript, to display in the browser.
If you are looking for tree structures to switch layers on and off and other more advanced UI widgets then GeoExtJS is worth looking into.
Server
If you are only displaying points then you can create dynamic KML or GeoJSON on your web server and reference this in OpenLayers. There are libraries in all languages that you can build on. Python has become a key languages in geospatial technologies, and has GeoJSON libraries you could script with.
GeoServer and MapServer are both server-side programs that allow you to serve out spatial data from databases with symbology, labelling etc. If you are having lots of different datasets then its worth spending time setting this up.
MapServer is written in C++ so it is probably easier for you to try this - however you may be able to get away without needing any specialist server side software, especially if your data consists of X,Y / Lon,Lat values that can be easily plotted as points.