I have noticed that different people look at different places when praying. Does it matter where we look when we pray and where should we really be looking at during each step?
Direction at which to look while praying: views and evidences
The direction to look at while praying is an Issue of madhab views.
AFAIK only the Maliki madhab recommends looking straight forward while praying (no matter where you are), so if one would be in the Masjid al-Haram one would be able to look at the Kaaba, as at least all madhabs and Muslims agree on facing the direction of Kaaba while praying. Their major evidence for this is
The majority of Sunni madhabs recommend looking at the place of sujud (prostration), some of them say while sitting for tashahud one should look at his finger. Sharyk al-Qadi even said: while standing one should look straight forward, while bowing to the place of prostration, while prostrating towards his nose and while sitting towards his lap. ...
The major evidence for them is a hadith from sahih ibn Hebban, see also for example in sunan an-Nasa-i and sunan abi Dawod. See also other references in this fatwa which considers the Maliki view as weak. See also this hadith which has according Imam a-Shawkani a hassan isnad (he added that the Narrator Musa ibn Abdulaj ibn abi Omayyah only appears in ibn Majah's hadith collection among the six major sunni hadit collection) and shows how people -apparently- changed their attitude while praying.
The most important on the direction to look at and how (open the eyes or shut them) while praying -and most scholars agree about that- is that one find a state of proper focus and humility (khoshu'), but there are a few things which have been clearly deemed like:
Looking at the Kaaba
There are a few ahadith which have been qualified as da'if like this one which has been compiled by at-Tabarni which seem to recommend looking towards a Kaaba and even pretending it is a worship. See also the hadith compiled by al-Bayhaqi in his sho'ab al-Iman:
النظرة إلى الكعبة عبادة
See also this fatwa in Arabic (a translation Urdu is also available)
Your question about the Angles seem to be related to this as the ahadith quoted by at-Tabarni (which have all untrustworthy narrators in their chains) count an amount of mercy which descends for the people who are praying, performing tawaf or looking (at the Kaaba). But doesn't really answer your question! But note we can't see angels so how would this have any influence on our or their prayers?
It is forbidden to lift one's eyes toward the sky in prayer. Abu Huraira reported Allah's Apostle saying "People should avoid lifting their eyes towards the sky while supplicating prayer, otherwise their eyes would be snatched away." (Bukhari vol. 1 no. 863)
With this you can get to know the essence of looking when you're praying. While praying, you're talking to God (one side communication, i.e. from your side). You need to understand that you need to be most humble, while directly talking to God. Hence different schools different thoughts. All emphasize that we lower our gaze in front of God. That is showing respect. There are no strict rules that you must look at a specific point only. Your not doing Yoga there. You're trying to communicate from the bottom of your heart.
So, the more humble you feel in front of God, the more your head bends down and down.
May the creator guide us all.
بسم الله الرحمن الرحیم
When you are standing up, you'd better see the place of your Sajdah. (Shia Muslims, look at the thing on which they Sajdah, such as Turbah)
When you are in Qunut situation, you'd better see the palm.
When you are in Roku', it is better to look at the place which it between your two feet.
When you are in Sajdah, it is better to see toward your nose.
When you are in Tashahod, it is better that you see to your skirt (or toward the tip of knees)
(Imam Baqer (a.s.) taught it to someone (for Roku', Sajdah, Tashahod))
In Salah we have to look concentrate at the place of sajdah but while tashahud we have to focus on our index finger (right hand)...:) for better understanding and authentic ahadiths refer to book "Sahih Namaz E Nabwi"... proper way of Salah according to sunnah you can find in this book in detail...:)