Html File Updatable?

I am currently in a website design class at my high school and have everything setup for my site. I uploaded all my html files and took a look at the site. After a few minutes of looking for problems I noticed that each browser (firefox & internet explorer) had minor problems each different. I believe that this problem is because the program i used at school (Dream weaver) was out of date. Is there any way i could update these html files so that the html files are compatible with the latest browsers?