Okay.First off,search function top right is your friend and will stop you pissing off the regulars in this forum with questions that have been covered before.Next you are probably looking at 3 different types of shoe.
Regards running shoes read this post I put up in another forum
"are you a pronator,suppinator or neutral runner?Will you need a support,cushioned or motion control shoe?Do you strike the floor with your heel or forefoot?I ,as someone who weighs in at 16 stone and lands heel 1st will require different shoes from someone who is 8 stone(damn whippet!!) and lands on his forefoot 1st....
Sorry if this is a bit obvious...
This is quite a good site to start with
http://www.runnersworld.co.uk/news/article.asp?UAN=481
Normally there will be a running specific shop near you and these will often have a gait analyser that can see how you run(sudden attack of 3 blind mice there?) and can recommend a pair of shoes.Have a look in the running mags for the local shop to you...
And I really hope you didnt know all this already.....
And this also explains quite well about the types of feet/shoes too
http://www.sportsshoes.com/misc.php...unningshoeforme
Dont be afraid to buy a pair of shoes that are discounted either as running companies often change their colourways and so retailers get left with surplus shoes they need to sell cheaply.Sometimes though its cos the shoe is crap....."
You need to work out what TYPE of running shoe before you buy something.There are different ypes for different people as explained above.
And what type of martial art do you do so people can recommend for that?
running shoes.. combat shoes... 
