The best surfaces to use for each activity (in no particular order) and the best shoes to wear when training on them:
1) Weightlifting – Rubber matting for proper footing and to lessen damage to flooring.
Not wearing shoes is not an option due to hygiene and safety reasons. Wear either very thin basic shoes that are just enough to protect from stubbing toes (no open toed shoes) or court/tennis/basketball shoes for greater support (but may provide more heel lift than some may want).
2) Sprinting/High Intensity Running Intervals – Grass or track for decreasing of “impact” forces when possible. Uphill running on cement or asphalt to increase push off challenge but decrease impact on landing.
Wear thin sprinting/spiked shoes if significant support not needed, otherwise wear running shoes which will slow you a bit, but provide better support.
3) Swimming – Just kidding.
4) Walking – Any surface. Cement for easier push off, but harder landing and grass for easier landing but harder push off.
Pool walking is not an option to improve the skill of walking or to gain bone density (land is required) and strength benefits of actual walking on land. Wear a stiffer, supportive shoe like a court/tennis shoe over a running shoe. Walking fast? More toward running shoes.
5) Jogging/Distance Running – Grass, sawdust trail, or track (cement or asphalt won’t necessarily hurt you, but proper running mechanics and footwear are thus essential).
A good basic running shoe that provides support and stability. There is nothing wrong with a good barefoot (or use of barefoot shoes) jog provided it is on grass or on the track, and with very gradual increases in duration and speed over time (be careful).
6) Jumping – Basketball court or group exercise room to increase jump height, or grass, track, loose dirt/sand to decrease landing forces (every benefit has some risk and vice versa).
Wear court/basketball shoes for optimal, more stiff, stability and support for this explosive activity.
7) Agility – Basketball court, group exercise floor, or turf (be cautious with real grass as surface may be loose and create slippage potential for injury and decrease ability to be agile).
Basketball or court shoes with proper lateral support. Running or barefoot shoes will dramatically increase the probability of rolling ankles and thus also increasing potential for knee, hip, or lower back injuries.