Sounds like your article links don't have actual menu associations, so Joomla opens them by default on the Home page. Use a Hidden Menu to associate the articles to menu items so they open on their own pages.
To control which modules display on which articles you have to manually set the Itemid portion (this represents the menu page the article resides on) of the link URL.
Go to Admin -> Menu's -> Menu Manager and then click on the NEW icon in the upper RH corner of the screen.
In the screen that opens, enter the following information:
Unique Name: hidden (or invisible, or whatever you want to call it - this should be a single word with no spaces or other special characters)
Title: Hidden Menu or whatever you want to call it
Description: Enter a descriptive phrase
Module Title: LEAVE BLANK
Do not create a "mod_mainmenu" module for the hidden menu. This menu will never be shown on your site, so no menu module is needed.
Once you click SAVE you will be returned to the Menu Manager screen.
Now, when you hover over the MENUS tab at the top of the Joomla screen, you should see your hidden menu listed there. Click on it, and then click on the NEW icon in the upper RH corner of the screen.
Here you will create your NEW Menu item. In the dialogue box that opens, select Menu Item Type: Articles > Single Article
In the modal window that opens SELECT the article that you want placed in the hidden menu. Make sure that the Menu Location field shows your hidden menu title. Fill in the other fields as you normally would. MAKE NOTE of the Article ID# in the farthest RIGHT column called "ID", we'll call ours "xx".
Next click on the SAVE icon in the upper RH corner of the screen.
When you are returned to the Menu Item Manager screen, make note of the Item ID that is assigned to your hidden menu item. The ID should be the farthest column on the right, we'll call ours "yy".
To create a link to the article, use the following format:
Replace xx with the ID number of the article you want to link, and replace yy with the ID number assigned to your hidden menu.
Now test the URL to make sure that it works.