Praise be to Allah.
The one who is able should not delay the regular Sunnah prayer of ‘Isha’ until after Taraweeh, because the time for Taraweeh begins after praying the two rak‘ahs [of Sunnah prayer].
It says in al-Mawsoo‘ah al-Fiqhiyyah (25/281):
With regard to Taraweeh prayer, the time for it begins after finishing the Sunnah prayer of ‘Isha’, and lasts until just before dawn by the amount of time it takes to pray Witr after having prayed Taraweeh.
This does not mean that praying Taraweeh without having done the Sunnah prayer of ‘Isha’ is invalid.
Shaykh Mansoor al-Bahooti (may Allah have mercy on him) said:
If a person prays Taraweeh after ‘Isha’ and before the Sunnah prayer of ‘Isha’, that is definitely valid, but it is better to do it (Taraweeh) after the Sunnah prayer, according to what is stated by the scholars.
End quote from Kashshaaf al-Qinaa‘ (1/426).
But is it valid for a Muslim to pray two rak‘ahs of Taraweeh with the intention of praying the Sunnah prayer of ‘Isha’?
The answer is: Yes, that is valid; in fact it is even valid to pray ‘Isha’ behind someone who is praying Taraweeh.
Shaykh Muhammad ibn Saalih al-‘Uthaymeen (may Allah have mercy on him) said:
You may join the imam in Taraweeh with the intention of praying the obligatory prayer, i.e., with the intention of praying ‘Isha’, then when he says the tasleem, get up and do two more rak‘ahs to complete the obligatory prayer, unless you are travelling, in which case you may say the tasleem with him, then join him in Taraweeh with the intention of doing the Sunnah prayer of ‘Isha’. If you are not travelling, then when you have done the Sunnah prayer of ‘Isha’, join the imam in Taraweeh. It does not matter if the intentions of the imam and of the person praying behind him are different; in other words, it is permissible for the imam to intend to offer a naafil (supererogatory) prayer and for the person praying behind him to intend to offer the obligatory prayer. This is what was stated by Imam Ahmad, that it is permissible for a person to pray ‘Isha’ behind someone who is praying Taraweeh.
End quote from ash-Sharh al-Mumti‘ ‘ala Zaad al-Mustaqni‘ (4/66).
But what he prays – in both cases – is not regarded as coming under the heading of praying qiyaam [voluntary prayers at night], because Taraweeh prayer is separate in and of itself, and cannot be put together with the Sunnah of ‘Isha’ with one intention (niyyah) – and this applies even more so to the obligatory prayer – rather he should intend to pray the Sunnah prayer of ‘Isha’ on its own, and his qiyaam will be lacking those two rak‘ahs.
Shaykh ‘Abd al-‘Azeez ibn Baaz (may Allah have mercy on him) said:
The Sunnah prayer of ‘Isha’ is a Sunnah mu’akkadah (confirmed Sunnah), and is two rak‘ahs. The Sunnah is to offer this prayer before Taraweeh, because it is a separate Sunnah, and Taraweeh is a separate Sunnah.
End quote from Fataawa as-Shaykh Ibn Baaz (30/56).
In order to attain the virtue of all of them, he should speak to the imam and ask him to wait, after the obligatory prayer of ‘Isha’, so that the worshippers can recite tasbeeh and dhikr, and offer the Sunnah prayer, then he can begin the Taraweeh prayer.
If the imam refuses to do that, or if the time is too short and it is not possible to offer the Sunnah prayer within that time, then there he has the choice:
Either he can delay the Sunnah prayer until after Taraweeh, so long as that is not after half way through the night, because that is when the time for ‘Isha’ and its Sunnah prayers ends;
Or he can offer the Sunnah prayer of ‘Isha’ in between the rak‘ahs of Taraweeh, during the time when the worshippers have a rest, or when a short talk is given. This is not included in the advice that some scholars give that it is not allowed to offer naafil prayer in the break in Taraweeh, because this prayer is raatibah [a regular Sunnah] and is not naafil in an absolute, unqualified sense.
Or he may pray the first two rak‘ahs of Taraweeh with the intention of praying the Sunnah prayer of ‘Isha’.
And Allah knows best.