Augustine proposed that "asking" refers (as you say) to prayer; that "seeking" refers to diligently searching for truth; and that only after having found the correct way we are prepared to "knock" in order to gain entrance. He explains:
For every one that asketh receiveth; and he that seeketh findeth; and to him that knocketh it shall be opened.” The asking refers to the obtaining by request soundness and strength of mind, so that we may be able to discharge those duties which are commanded; the seeking, on the other hand, refers to the finding of the truth. For inasmuch as the blessed life is summed up in action and knowledge, action wishes for itself a supply of strength, contemplation desiderates that matters should be made clear: of these therefore the first is to be asked, the second is to be sought; so that the one may be given, the other found. But knowledge in this life belongs rather to the way than to the possession itself: but whoever has found the true way, will arrive at the possession itself which, however, is opened to him that knocks.
In order, therefore, that these three things—viz. asking, seeking, knocking—may be made clear, let us suppose, for example, the case of one weak in his limbs, who cannot walk: in the first place, he is to be healed and strengthened so as to be able to walk; and to this refers the expression He has used, “Ask.” But what advantage is it that he is now able to walk, or even run, if he should go astray by devious paths? A second thing therefore is, that he should find the road that leads to the place at which he wishes to arrive; and when he has kept that road, and arrived at the very place where he wishes to dwell, if he find it closed, it will be of no use either that he has been able to walk, or that he has walked and arrived, unless it be opened to him; to this, therefore, the expression refers which has been used, “Knock.”1
There is also an interesting subtlety in the Greek here that is impossible to bring out without being overly literal. A simple imperative to do something is normally indicated with the verb in the aorist tense. When the verb is in the present imperative - which is the case in the above verses - a continuing action is indicated.2 A more literal translation of Matthew 7:7-8 would be:
Keep on asking, and it shall be given to you; keep on seeking, and ye shall find; keep on knocking, and it shall be opened to you.3
This aspect of the passage - that of the need of persistence in the believer - was recognized by Greek commentators on this passage. Theophylact (Byzantine, 11th cent.), for example, wrote:
In what has preceded the Lord has commanded us to do great and difficult things. Here he shows hus how these things can be accomplished: through unceasing prayer. For He said, "Ask", that is, "keep asking," meaning "ask continuously". For He did not say, "Ask one time."4
This sense also comes out in the (Greek) commentary of John Chrysostom (4th c.):
However, not simply to ask did He command us, but with much assiduity and earnestness. For this is the meaning of “seek.” For so he that seeks, putting all things out of his mind, is taken up with that alone which is sought, and forms no idea of any of the persons present. And this which I am saying they know, as many as have lost either gold, or servants, and are seeking diligently after them.
By “seeking,” then, He declared this; by “knocking,” that we approach with earnestness and a glowing mind.
Despond not therefore, O man, nor show less of zeal about virtue, than they do of desire for wealth. For things of that kind thou hast often sought and not found, but nevertheless, though thou know this, that thou art not sure to find them, thou puttest in motion every mode of search; but here, although having a promise that thou wilt surely receive, thou dost not show even the smallest part of that earnestness. And if thou dost not receive straightway, do not even thus despair. For to this end He said, “knock,” to signify that even if He should not straightway open the door, we are to continue there.5
1 Sermon on the Mount II.XXI.71-72
2 "The aorist imperative predominates in the New Testament. The distinction between the present and the aorist imperative seems to be ignored by many translators. We are safest when we assume that the inspired author had a reason in his mind for using one rather than the other. The aorist imperative views the action from the unitary perspective. It is an action that is either transient or instantaneous or to be undertaken at once ... Every time Jesus asked a person to follow Him, He used the verb ἀκολουθέω in the present imperative, that is 'keep on following Me' ... This denotes continuous or repeated action; that is, 'Begin now to follow Me, and continue doing so'" (The Orthodox New Testament Praxapostolos, p.609; see also Wenhem, The Elements of New Testament Greek, p.74.).
3 Orthodox New Testament Evangelistarion
4 The Explanation by Blessed Theophylact of the Holy Gospel According to St. Matthew, p.64
5 Homilies on Matthew XXIII.4