Your total Stealth mod is +8.
Your mistake here is that you are adding your proficiency with Dexterity saving throws to your total. This is incorrect. Saving throws are a special kind of die roll, separate from the (related) ability check and attack roll. Saving throws are generally reactionary; the DM will ask you to make one in response to certain kinds of threats. Skills are used proactively, that is, generally the DM will call for a ability check when you are trying to accomplish something.
Stealth is a skill. Thus, to calculate the corresponding modifier for Stealth checks, you will apply the related ability modifier (for Stealth, the corresponding ability is Dexterity; your Dex mod is +4, as you stated) and your proficiency bonus (if you are proficient; at level 1, this is +2). Since you chose Stealth as one of the skills you have Expertise in, this last number is doubled.
Thus, when you make a Stealth check, you roll a d20 and add:
+4 (Dex) + [+2 (Proficient) * 2 (Expertise)]
for a total of +8.