Tabearuki (according to blog.japundit.com/archives/2006/04/20/2341/) means something like trying a variety of dishes from a variety of places.

So: shumi wa tabearuki desu means 'my hobby is eating many foods at many places'
and oishiimono wo tabemasu means 'Eat delicious things'

Hope this helps!

I am still new to learning Japanese and have bitten off a little more than I can handle. I have a pen pal I have been talking to in Japan (he doesn't speak english). This is the third time out of 26 emails I have found my self here asking for help. Please bear with me if you can. I asked him what his hobbies were. I can't be sure what exactly he says here, I think it has to do with the food carts in japan which if I remember they call walking food, but I could be totally wrong so I am asking for your guys' help so I don't sound weird when I write my reply.

Shumiwa tabearukidesu
oisiimonowo tabemasu


Please keep in mind this is exactly how he wrote it I double checked. But I know from experience that it is slang and isn't exactly how it should be. Sorry in advance for being reliant. But please know that I am very grateful for all the help I receive here.

Saiyuri

Thank you so much. That did help out.^_^