Praise be to Allah.
The dhikr after salaah referred to in the question has been reported in a hadeeth reported by Abu Hurayrah from the Messenger of Allaah (peace and blessings of Allaah be upon him): Whoever glorifies Allaah (says Subhaan Allaah) thirty-three times immediately after each prayer, and praises Allaah (says Al-hamdu Lillaah) thirty-three times, and magnifies Allaah (says Allaahu akbar) thirty-three times, this makes ninety-nine, then to complete one hundred says Laa ilaaha ill-Allaah wahdahu laa shareeka lahu, lahul-mulk wa lahul-hamd wa huwa ala kulli shayin qadeer (There is no god except Allaah Alone, with no partner, His is the power and His is the praise, and He is Able to do all things) his sins will be forgiven even if they are like the foam of the sea.
(Reported by Muslim, 939).
A number of different descriptions of the dhikr after salaah have been narrated, an example of which has been referred to in the question. The evidence for this is the hadeeth of Abd-Allaah ibn Amr (may Allaah be pleased with him) who said: The Messenger of Allaah (peace and blessings of Allaah be upon him) said: There are two qualities, no Muslim man acquires them but he will enter Paradise, and they are simple and easy. He should glorify Allaah (say Subhaan Allaah) ten times immediately after each prayer, and praise Him (say Al-hamdu Lillaah) ten times and magnify Him (say Allaahu akbar) ten times. I saw the Messenger of Allaah (peace and blessings of Allaah be upon him) counting this on his fingers. He said: That makes one hundred and fifty on the tongue, and one thousand five hundred (hasanaat) in the scales.
[Translators note: each of three phrases repeated ten times makes thirty; multiplied by the number of daily prayers, which is five, makes one hundred and fifty. Each of these good deeds of the tongue will be rewarded with ten hasanaat which will be added to the total of good deeds to be weighed in the balance or scales on the Day of Judgement]. When you go to bed, glorify Him and praise Him and magnify Him one hundred times: that will be one hundred on the tongue and a thousand in the scales. Who among you does two thousand and five hundred sayiaat (bad deeds) in one day? They said: How could we not count (our sins)? He said: The Shaytaan comes to any one of you whilst he is praying and says, Remember this, remember that, until he finishes his prayer and does not do (this dhikr), or he comes to him when he is lying down and makes him sleepy, until he falls asleep (without doing this dhikr). (Abu Eesaa said: This is a hasan saheeh hadeeth. Sunan al-Tirmidhi, 3332).
Among the adhkaar that may be recited before going to sleep, as referred to in the question, is that reported in the hadeeth of Ali ibn Abi Taalib (may Allaah be pleased with him), who said that Faatimah (upon whom be peace) came to the Prophet (peace and blessings of Allaah be upon him) and asked him for a servant. He said, Shall I not tell you of something that is better for you than that? When you go to sleep, say Subhaan Allaah thirty-three times, Al-hamdu Lillah thirty-three times, and Allaahu akbar thirty four times.
(Reported by al-Bukhaari, 4943).
As regards counting this tasbeeh, the Sunnah is to use the fingers of the right hand, although it is permitted to use the left hand. The evidence that the right hand is preferable was reported by Abu Dawood (may Allaah have mercy on him), who said: Ubayd-Allaah ibn Umar ibn Maysarah and Muhammad ibn Qudaamah, among others, told us: Aththaam told us from al-Amash from Ataa ibn al-Saaib from his father from Abd-Allaah ibn Amr who said: I saw the Messenger of Allaah (peace and blessings of Allaah be upon him) counting the tasbeeh. Ibn Qudaamah said: With his right hand.
(Sunan Abi Dawood, Kitaab al-Salaat, Baab al-Tasbeeh bil-hasaa).
It was said in Haashiyat al-Tahtaawi: It is correct that he (peace and blessings of Allaah be upon him) used to count the tasbeeh on his right hand. It was reported that he said: Count with your finger tips, for they will be asked and they will be made to speak. Ibn Hijr said: Counting the tasbeeh with the fingertips is better than using the subhah (prayer-beads or rosary).
The sunnah is to count with the fingertips by putting them on the palm of the hand. As to precisely which finger of the right hand one should start with, I have no specific information on this.
And Allaah knows best.