First of all, I apologize for the long post. If you just want to get the answer do not read the text below.
I understand Azeri very well, lets say, %95 of what they say and almost everything in the writen form. The Azerbaijanis also understand us very well. However I don’t think that someday I will speak Azeri because Azeri and Turkish are so close. In the eastern part of Turkey people talk Turkish like Azeris. An examples:
Toprak(Turkish) - Torpak(Azeri) - Torpak (in the eastern part of the country) / soil, ground, land
I don’t know why but saying “torpak” sounds “funny”. (The fact is only the farmers in Turkey say torpak and it sounds a little bit “different”)
Btw Azeri is actually the second biggest Turkic language because in Iran there are app. 30 million native Azeri speakers, one of them is Ayetullah Khamei. This video shows the conversation between Ataturk and Shah in Turkish-Azeri, if you are interested in there is a link here.
I am not sure, if Azeri and Turkish are two different languages. I think, they are not.
I can understand Gagausian perfectly and %95 of Crimean Tatar. (Still I am not sure, if they are seperate languages. My answer is, no they are not.) I have never studied them. I only listen to the Youtube channels, football matches and news in these languages.
THE ANSWER
Uzbek would be a good start. They use the latin alphabeth and there are app. 35 million native Uzbek speakers. The words are not very different.
My second choice would be Kazakh. They use the cyrillic alphabeth and their words are a little bit different than ours. Whenever I hear Kazakh, I feel like, I am listening to the grandsons of the mighty Khans. 
Kazakhs use j and ş instead of y and ç. (j- french j, y like in York, ş is sh and ç is ch)
Yıldız (Tur) - Jıldız (Kaz) - Star
Kazakça (Tur) - Kazakşa (Kaz) - Kazakh language
EDIT
Here there is a video of Ayetullah Xamenei, the supreme leader of Iran.
Translation from Azeri
Some Farsi words… At 0.10 He starts to speak Azeri.
“In the Islamic Republic of Iran, the Turks and the Persians are equal. (Have equal rights) Together…” (applause) “Let me continue…”
“In the Islamic Republic of Iran, the Turks and the Persians are equal, they are Iranian, Muslim and brothers.”
Trancription
İran-i İslami’de Türk de Fars kimin, Fars da Türk kimin, birbiriynen… goyun cümle tamam olsun… Arz edirem, İran-i İslami’de Fars da Türk kimin, Türk de Fars kimin, hami İranidi, muselmandi, birbiriynen gardaşdi.