Once you've downloaded your datasets there's a wide range of different tools available to help you query, convert, and analyse the data.

Quick visualisation and data format conversion

• GeoJSON.io: A simple website for that allows for uploading, converting, and visualising moderately sized spatial datasets. You can also draw and create your own spatial datasets - handy for creating dummy datasets!
• Ogre: A handy website to convert many common spatial data formats (including shapefiles) to the modern GeoJSON format.
• NationalMap: NationalMap is not only a handy catalogue of government datasets, you can also upload and visualise other datasets on it (supports common formats like GeoJSON, KML, GPX, and more).

Platform as a Service Maps

• Carto (formerly CartoDB): A simple, easy-to-use platform for everyone (not just geospatial nerds). Upload your data, hook into common datasets (e.g. country borders, administrative boundaries), and create beautiful web and mobile maps right from your browser.
• ArcGIS Online: Make and share beautiful maps, and do everything in between. Maps, apps, analytics, administration, collaboration through an easy-to-use mapping solution.
• MapBox: Mapbox is a mapping platform for developers. Easily integrate location into any mobile or online application. Search, geocoding, real-time data, directions and routing, 2.5D and 3D maps.

Desktop software

• QGIS: The open source Geographic Information System. Create, edit, visualise, analyse, and publish geospatial information on Windows, Mac, Linux, and BSD.
• ArcGIS Pro: ArcGIS Pro reinvents desktop GIS. Design and edit in 2D and 3D, work with multiple displays and layouts, and publish finished web maps directly to ArcGIS Online or Portal for ArcGIS, connecting you to users throughout the world.

Databases

PostGIS: Spatial databases aren't just for storing data - they're great for conduct complex analysis tasks if you're happy writing SQL queries. For spatial databases consider nothing but the best: the PostGIS extension to PostgreSQL. If there's a spatial query or manipulation you need to do then PostGIS has it; and hundreds of other functions besides. It's easy to get PostgreSQL + PostGIS running quickly on macOS, Windows, or AWS.

Command-line

GDAL: Check out GDAL (Geospatial Data Abstraction Library), which has readers and writers for over 50 types of spatial data. GDAL bindings also exist in Python, .NET, et cetera.