Luckily, and unlike most states, Delaware has a free online system for checking legal entity name availability. That system can be found here.
It is relatively easy to use and will make your filing process more efficient. For example, if you file a certificate of incorporation with a name that is already taken, your filing will be rejected.
If the name you want is available, you will be given the option to reserve this name for $75. Note that this reservation expires and does not last indefinitely. Should you do this? That depends.
If you are going to file for the entity yourself, and you know without a doubt that this is the only name that you want, perhaps you could benefit from reserving the name. However, if you are going to use a law firm or other third party to file your entity, it will be difficult for them to submit the filing using the name that you had already reserved (yes, this makes no sense and is a weird quirk in the Delaware system). So, keep that in mind.
Also, it is important to note that just because a name is available for reservation in Delaware does not mean that it is available in other states, or that name availability also means that a trademark check has been done, or that the domain is available.
The Delaware online system only checks if there is another entity either formed or registered in Delaware with the same name (or a name that is very similar which may prohibit you from using the name you want). Therefore, before you file with a certain name you should perhaps do a quick Google search to see what it brings up. While Google isn’t definitive, it may alert you to potential issues even if the name is available with Delaware. It would also be helpful to check for any potential conflicting trademarks with the USPTO.