Meaning of SubhanAllah wa Bihamdihi
Question

What are the deeper meanings of subhanallah wa bihamdihi and bismillah? Please explain, for they will help me in my prayer.
Praise be to Allah.Meaning of Subhan Allah
The words of tasbih, “Subhan Allah”, incorporate one of the greatest fundamental meanings of Tawhid and one of the basic principles of belief in Allah , may He be glorified and exalted, which is declaring Him to be above any fault, shortcoming, corrupt notions or false idea.  
The basic linguistic meaning points to this and is taken from the word al-sabh which means distance or being far from something. 
Al-‘Allamah Ibn Faris said: “The Arabs say: Subhana min kadha meaning How far away is such and such. Al-A’sha said: 
Subhana min ‘Alqamat al-fakhir aqulu lamma jaani fakhruhu
How surprising is ‘Alqamah when he boasts; I said that when I heard about his boasting. 
Some people interpreted these lines as meaning: ``It is very surprising when ‘Alqamah boasts, and this is close to the meaning of distance, because the speaker is far away from pride.” (Mu’jam Maqayis al-Lughah, 3/96)
So tasbih of Allah means declaring that one’s heart and mind are far removed from thinking that there is any shortcoming in Him, or attributing any evil to Him, and declaring Him to be far above any fault that is attributed to Him by the mushrikin and heretics. 
It is this meaning that is indicated in the Quran. 
Allah says (interpretation of the meaning): 
“No son (or offspring) did Allah beget, nor is there any ilâh (god) along with Him. (If there had been many gods), then each god would have taken away what he had created, and some would have tried to overcome others! Glorified be Allah [Subhan Allah] above all that they attribute to Him!” [al-Muminun 23:91] 
“And they have invented a kinship between Him and the jinn, but the jinn know well that they have indeed to appear (before Him) (i.e. they will be called to account). Glorified be Allah [Subhan Allah]! (He is free) from what they attribute unto Him!” [al-Saffat 37:158-159]
“He is Allah, beside Whom La ilaha illa Huwa (none has the right to be worshipped but He), the King, the Holy, the One Free from all defects, the Giver of security, the Watcher over His creatures, the All-Mighty, the Compeller, the Supreme. Glory be to Allah! (High is He) above all that they associate as partners with Him.” [al-Hashr 59:23]
It also appears in the report narrated by Imam Ahmad in al-Musnad (5384) from Hudhayfah (may Allah be pleased with him), describing the recitation of the Prophet (peace and blessings of Allah be upon him) in the night prayer. He said: “When he came to a verse which spoke of Allah being above any fault or defect, he said Subhan Allah.” (Classed as sahih by al-Albani in Sahih al-Jami’ (4782) and by the editors of al-Musnad.) 
Imam al-Tabarani narrated in his book al-Du’a a number of reports explaining this meaning of this word, which he compiled in a chapter entitled Tafsir al-Tasbih (explanation of the phrase Subhan Allah) (p. 498-500). Among the reports that he quoted in this chapter are the following: 
“It was narrated from Ibn 'Abbas (may Allah be pleased with him): 
Subhan Allah means declaring Allah to be above every bad thing. 
It was narrated that Yazid ibn al-Asamm said: 
A man came to Ibn 'Abbas (may Allah be pleased with him) and said: 
“La ilaha ill-Allah ” we know: there is no god other than Him. “Al-hamdu Lillah” we know: all blessings come from Him and He is the One Who to be praised for them. “Allahu akbar” we know: there is nothing greater than Him. But what does “Subhan Allah” mean? 
He said: This is a word that Allah has approved for Himself, and has enjoined His angels to say, and inspired the elite of His creation to utter. 
It was narrated that ‘Abd-Allah ibn Buraydah narrated that a man asked ‘Ali (may Allah be pleased with him) about the phrase “Subhan Allah” and he said: It is veneration of the majesty of Allah. 
It was narrated that Mujahid said: 
Tasbih means to declare that Allah is free of anything bad. 
It was narrated that Maymun ibn Mahram said: 
Subhan Allah is veneration of Allah and is a phrase by which Allah is venerated.  
It was narrated that al-Hasan said: 
Subhan Allah is a restricted phrase that cannot be used with reference to any of creation. 
It was narrated that Abu ‘Ubaydah Mu’ammar ibn al-Mathanna said: 
Subhan Allah is declaring Allah to be above (all defects). 
Al-Tabarani said: al-Fadl ibn al-Habbab told us: I heard Ibn ‘Aishah say: 
When the Arabs find something objectionable and regard it as very serious, they say “Subhan”. It is as if it is declaring Allah to be above every bad thing that it is inappropriate to attribute to Him. 
Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said: 
“The command to glorify Him (by saying Subhan Allah) also implies declaring Him to be above every fault and shortcoming, and affirming the attributes of perfection for Him. Tasbih implies declaring Him to be above any fault and venerating Him, and veneration implies affirming praiseworthy attributes for which He is praised, and that implies declaring Him to be above any faults, praising Him, magnifying Him and affirming His Oneness.” (Majmu’ al-Fatawa 16/125) 
Meaning of Wa Bihamdih
With regard to the meaning of the phrase wa bi hamdihi: in brief, it means combining tasbih (glorification) and hamd (praise), either simultaneously or consecutively. So the meaning is: I glorify Allah as I am praising Him or I glorify Allah and I praise Him. 
Al-Hafiz Ibn Hajar (may Allah have mercy on him) said: 
“With regard to the words “wa bi hamdihi”: 
It was said that the waw indicates simultaneous action, so the meaning is: I glorify Allah at the time when I am praising Him for His help and support. And it was said that it refers to consecutive actions, so the meaning is: I glorify Allah and I praise Him. 
It may be that the meaning is: I laud Him by praising Him. So “Subhan Allah” is a separate phrase, and “bi hamdihi” is a separate phrase. 
Al-Khattabi said concerning the phrase “Subhanaka Allahumma Rabbana wa bi hamdika (Glory be to You, O Allah, our Lord, and praise) – i.e., by means of Your strength which is a blessing for which You are to be praised I glorify You, not by my power and strength.” (Fath al-Bari, 13/541; see also al-Nihayah fi Gharib al-Hadith by Ibn al-Athir, 1/457) 
Meaning of Bismillah
With regard to the meaning of the Basmalah, “Bismillah (in the name of Allah)”, this has been explained and discussed in the answer to question no. 21722 . 
And Allah knows best.