If you can maintain focus on researching one family member at a time, you will have more sources to help you confirm (or disprove) those ancestors you have very little information about.
Have a Strategy
Create a plan for collecting information about an ancestor. Have a look at your paper and pencil pedigree, and see what is missing.
One possible strategy would be to start with your father, and follow along the numerical reference numbers of each family tree member, collecting (and recording!) as much data as you can find for each ancestor before moving on to the next.
Or you could work along one particular line, perhaps researching each ancestor along your paternal line, or maternal line first. Just make sure you spend time working on each member of your family tree so that nothing gets overlooked.
If you have developed a strategy, you already know which ancestor you want to start with. Now you need to search all of your available online resources (we'll get to genealogy libraries and other repositories when we have enough data to know what we are looking for).
I like to start with FamilySearch because it is free and has an extensive collection of Texas family history resources. Enter the data you know about your relative, and see what comes up.
If you don't find anything, go for something more specific, such as a marriage record. If you know the state where your ancestor was likely married, enter that in the marriage location. Remove the first names if you are still not getting any results. Be sure to add the spouse's last name, if you know it.
If you know the names of the ancestor's parents, search for birth records or indexes by searching the parents names and location. If that doesn't work, try just searching the parent's last names.
If the state's death certificates have been indexed, you can also find data by searching just for the parents' names or last names. This is a great way to find the children of an ancestor who's children would have died within a specific state during the period in which death certificates are currently available to the public.
For census records, search the name and residence location during a given census year. If you don't know where the ancestor lived, leave the location blank and look for clues (spouse, children) that will help you find the correct family unit.
Be sure to check Find A Grave to see if photos or information about an ancestor's grave site are available.
Save all scanned images you find, and copy textual data into a notepad or word pad text file along with the source information. Be sure to give all files a name that will make the data easy to find again.
Also write down your new findings on your pedigree and family group records.
Don't Get Discouraged
Some relatives are just very difficult to find much information on. If you've spent the whole day searching online databases, to no avail, it's okay. Move on to another ancestor. You can come back to that one later, perhaps when you are prepared to visit a genealogy library or local history archive.
Please feel free to share your own tips and strategies for online research of ancestors.