Hi,
if you want to connect to an analog phone line you need a line interface circuit. There exists dedicated IC's for that but often it is easier, especially for beginners, to rip apart a functioning telephone. You can then replace the hook switch by a relay and input your audio directly into the microphone using a small speaker, maybe from a headphone - in the end you build something like an acoustic coupler. If you generate DTMF tones using some library you should be able to dial. If you are more advanced into electronics you can try to replace the mic and speaker in the phone by audio transformers or opto couplers to make the connections to you Arduino. About a decade ago I build a alarm system that way that in event of an alarm starts dialing numbers and even provides a voice announcement.