First, I am not a Hebrew or biblical scholar. What I am, though, is an adult student who found an interest for the Hebrew language. After trying to teach myself Hebrew, then paying for online lessons, and then taking college level Hebrew courses, I think I have at least a good understanding of the best way to get started.
Second, if you can afford it, one of the best ways to learn Hebrew is to take lessons from an actual Hebrew speaking Israeli. That is if you want to read, write, and speak native Hebrew. Pronunciation is always an important element to learning a foreign language. If you just want to read Hebrew, for the sake of reading the scripture, modern teachers are still a good resource but don't forget the rabbis and biblical scholars who are also knowledgeable in the area. Likewise, you can find many free audio files as well as websites that will help you with pronunciation if you want to go at your own pace. I'll provide more links to resources later as we get started in the tutorials.
Third, Hebrew is a very concrete language that makes it rather easy to learn in my opinion. The hardest part is in the beginning, which includes learning an new alpha-bet, commonly called the aleph-bet in Hebrew, and reading from right to left. After you nail down many of the basics you find that Hebrew has a lot of shortcuts that make learning the language a lot simpler than something like English.
Fourth, the best way I have found to learn Hebrew is by learning the block script, what you will read in, the handwriting script, what you will write in, and the vowel marks, which helps with pronunciation. Learning all three and practicing them in the beginning really saves time latyer on. You don't necessarily need to learn the handwriting script if you are not interested in writing modern Hebrew, or learning the "niqud" vowel marks if you are just wanting to learn biblical Hebrew but I still encourage you to do so.
The image below shows you the Hebrew aleph-bet in block script. You can think of the block script as what you will use when reading Hebrew. It will be used in most printed materials and you can see how it has a calligraphy style appearance to it. Although the scriptures and other materials still use some variation of the block script, it is not as practical for writing in unless you are going to make a living writing Torah scrolls. Remember to read from right to left.
After looking at the block script hopefully you can see why the Hebrew came up with an easier way of writing the aleph-bet called the handwriting script. Although the letters are not connected like in cursive English, the handwriting script serves a similar purpose that allows modern speakers to write quickly in Hebrew. In modern Hebrew, you will use the script for writing and reading someone else's writing such as a letter etc.
The vowel marks "niqud" are depicted below. They are a series of dots and dashes that help guide the reader in pronouncing the words correctly. Don't worry. It looks like a lot to learn but many of the vowel marks are shortened forms of longer vowel marks. They are really easy to learn as long as you don't confuse them with English vowels, in the sense they mean anything standing alone like - A or I etc. The vowel mark will appear below, to the left, or to the write of the Hebrew character. The dashed circle in the images below depict where Hebrew a character would normally appear. Do not try to learn these at this point as they are in no particular order.
