Font licensing is a big deal. As I stated previously, this is something that is covered in a general IT class, in my country. While it has been some time since I took a "general" IT class, the general idea is to make sure you do have permission to use the font.
It is important to note that I am not a lawyer, and not even a lawyer can really tell you if you will be sued. While this depends on location, you can really be sued for anything, provided the claimant is confident enough in their case to warrant the required expenditure. For this reason, it may not necessarily be a good idea to go off the consensus of the team. As Alexandre states, you may ultimately want to contact a copyright lawyer, for peace of mind.
I'm noticing that license agreements for fonts are pretty varied
It depends where you look. A quick google search lead me to an article at Design Shack, which gives a very good overview of font licenses, and the need to acquire them.
My main concern is with a license that may not cover our intended uses, and lead to penalties.
In general, you are looking for a commercial license. From what you tell us, your game is not currently in the commercial stage, but you think that you might make it commercial at some point. Consider the implications of making your game commercial, where you do not have commercial license to use the font.
If you do not "upgrade" your license, in this scenario, you lead yourself to further legal drama. You are using a font outside of its license agreement. If you intend to "upgrade" your license, you are going to have to chase up the original creator/publisher, and arrange a new license. What if you can not get in contact with the original creator/publisher, or they simply refuse, which they are completely within their right to do? You would have to start over, and find a completely new font.
Where should I look for these fonts?
1001 Free Fonts offers a vast collection of fonts advertised as "free". I would not assume commercial license, but many do also offer the option to explicitly purchase a commercial license.
Google Fonts may be an option; rather than directly selling you fonts, Google offers an API service that provides access to a broad range of open source fonts. Google explicitly states permit-able use for commercial use and applications, but I can not work out if it would be feasible to integrate this into your game.
MyFonts is another site that explicitly states license to use the fonts in mobile applications. Across ten seperate fonts, every one of them offered a "pay once per unique application" license, and the fonts were far better priced than on fonts.com.
Another user on a closed question suggests checking out The League of Moveable Type, a site that focuses on free, open-source fonts.
CheapProFonts seems to be another suitable option. We have already had a user ask about using fonts from these guys - and a company representative actually answered, directly, to confirm that the one license was adequate for use in his game.
Are there any other options?
As already stated, you have the option of simply contacting the creator and asking for permission. You would be surprised how many authors would post a font as "free", and not even consider the legal implications that their font may not actually be "free". This process should take you about five minutes, and if you do not end up using the font, there is no real penalty. Just make sure that if you do use the font, you respect any requests such as acknowledgement.
There is another suitable option; one that will not cost you any money, or risk any foreseeable legal drama: Make it yourself.
Fonts can be rather complex, if you consider all of the styles and variations. If you can get away with defining your exact requirements, you may consider simply creating your own font. This is an option I have seen developers take, in the past. I have even seen developers use "creative license" to effectively hand draw the font they wished to use, based off copyrighted work that was outside of their budget. The new version could be considered their own work, and was therefore legally1 acceptable.
1 Note that legal is not the same as ethical; ultimately, you make that additional call on whether you consider this to be appropriate behavior. I would speculate that such action is only pseudo-legal, in that it would be next to impossible to actually proove, in court.