I have recently written a couple of short fiction stories and I would like to get them published. My searches on the internet has left me dazzled. There are so many option and most of them are just not what I'm looking for. Do you know a place where they publish short fiction or poetry? I'm not looking for a journal/magazine that you need to pay to get considered for publication or one where you have to wait for months to get a reply. I'm even thinking of starting my own journal/magazine for people just like me.
You don't mention having any other writing experience or what you may have done in terms of edits, etc., so my comments are assuming these are your first stories. Just because you've written some stories, it does not mean they are ready for publication. You may have great stuff there (evaluate them against some of the favorites that you've read) or they might not be ready for prime time yet. You may want to try workshopping them through a site like Scribophile.com, which is a good place to give and get critiques, or with a local writers' group. Making sure your writing is the best it can be will make the process of submitting to journals less painful, because you are more likely to be accepted.
If/when the stories are ready to send out, in addition to @what's suggestion about looking at the magazines you read, you can also take a look at Duotrope.com. They have a database you can search and links to the different journals so you can check the individual magazine's submission guidelines. Always follow the guidelines or the odds are your stuff will get tossed. Duotrope usually gives an indication of how long it takes to get a response (I can't remember, you may have to subscribe to get those statistics.)
Good luck. Keep writing. Work the craft.
"I'm even thinking of starting my own journal/magazine for people just like me."
Many journals make serious efforts to be respectable towards their contributors. They don't WANT to make you wait long, they just don't have the funding or time to answer everybody in a timely manner, let alone with good news.
Start your journal, and you'll see. You get a thousand entries in a week, at say an average word count of 3,000 words. That's 3 million words in a week. Good luck.