Assuming you are not on wifi, if you can surf to web sites in Safari and send and receive email, then you are connected to the data side of whatever network your SIM card connects to, so yes, I'd say you are good to go.
Most GSM carriers do not allow editing the APN settings (my AT&T iPhone 4 has no way to access those settings either) as they are provisioned from the SIM once it connects the device to the carriers network. If you have troubel with data, you can call the carriers support number, and if needed, the can push APN changes to your phone over the air.
Or you could try this - http://support.apple.com/kb/HT4839?viewlocale=en_US&locale=en_US
Jailbreaking will void your warranty, as well as the fact that you will no longer get any support from folks here.
Straighttalk is a MNVO (Virtual Carrier) of AT&T so this is why Straighttalk SIMs work for non unlocked AT&T iPhones .. BUT.. when you use a Straighttalk SIM the AT&T APN Settings are used and are hidden away for you to edit.
You can use the the iPhone Configuration Utility and use it to change the APN Settings or you you go to http://www.unlockit.co.nz (Whilst Connected to WIFI) and choose United States/Straightalk and this will set the APN Settings.
www.unlockit.co.nz uses the same method as the iPhone Configuration Utility so does not jailbreak or do anything which will void your Warrenty.