Under what conditions can one experience the power of the Holy Ghost? In the LDS Church, there is what is known as the power of the Holy Ghost, and the gift of the Holy Ghost (not to be confused with the gifts of the Holy Ghost).

The power of the Holy Ghost is when some truth or principle is taught or testified to you by the Holy Ghost, or, in other words, it is His authority to testify. For example, the Holy Ghost can testify that the Jesus is the Christ, or that the Bible is the word of God. It often comes when we ask God questions through earnest prayer.

The gift of the Holy Ghost is the right to have the Holy Ghost as your constant companion at all times and in all places, as you remain worthy. It is received by the laying on of hands.

Under what conditions can one experience the power of the Holy Ghost?

(Ref: https://www.mormon.org/beliefs/holy-spirit)
Generally speaking, the "power of the Holy Ghost" is simply the manner in which He accomplishes His mission.  Again, generally speaking, LDS members don't distinguish a difference between the "power of the Holy Ghost" or any other of His duties or blessings.  So, let's explore a moment.

1) You mentioned the "Gifts of the Spirit" such as tongues, healing, prophecy, etc.  Certainly these occur through the power of the Holy Ghost, but they are an inexclusive application of that power.  Indeed, though personal unrighteousness may bury the gift or gifts, we believe that everyone has at least one gift of the Spirit, no matter who they are or what they believe (D&amp;C 46:11).  Since this affects everyone and is conditioned only on personal righteousness, we'll set this aside as insufficient to answer your question.

Condition(s): Personal Righteousness

2) You mentioned the Gift of the Holy Ghost.  This isn't a specific power, but the ability (assuming once again personal righteousness) to have the Holy Ghost as a constant companion.  Let's start with Acts 2:38:


  Then Peter said unto them, Repent, and be baptized every one of you in
  the name of Jesus Christ for the remission of sins, and ye shall
  receive the gift of the Holy Ghost.


and end with D&amp;C 20:26:


  ...but all those from the beginning, even as many as were before he
  came, who believed in the words of the holy prophets, who spake as
  they were inspired by the gift of the Holy Ghost....


From the LDS perspective, before Baptism and giving the formal Gift of the Holy Ghost people can be inspired by Him (He primarily acting in His role as testator), but they are not going to receive more from Him.  No revelations, no prophecy, no second comforter, no spirit of promise.  The goal of the Holy Ghost before Baptism is to lead people (a) to do good and (b) to Baptism.  Now, this technically answers your question about the Power of the Holy Ghost, but let's delve a bit more.

Condition(s): personal righteousness, baptism by proper authority

(Authority is a discussion all its own.  In our belief, Baptism can only be authoritatively performed by an ordained priest of the Aaronic Priesthood in the LDS Church or any member of the Melchizedek Priesthood.  We don't recognize any other baptisms.)

3) The Holy Ghost has specific roles.  The scriptures I list below are by no means exhaustive.


(a) A teacher, bringing all things to rememberance.  (John
14:26) 
(b) A witness or testator (Acts 5:32)
(c) A motivator (Ezek 36:27)
(d) A conscience (Matt 12:18)
(e) A provider of revelation or visions (John 16:13)
(f) A sealer, the Holy Spirit of Promise (D&amp;C 76:53-54 (132:7))
(g) And, a Servant of God, doing as God requires (Moses 6:64)


All these things are done with the Power of the Holy Ghost, but most won't happen consistently unless given the Gift of the Holy Ghost.

(a-e) Conditions: personal righteousness &amp; authorized baptism.

(f) and (g) are specific issues and rare.

(f) The Holy Spirit of Promise (i) seals a person's calling and election sure (2 Pet 1:10) and (ii) seals a temple marriage for time and all eternity.  Both are required for the Celestial Kingdom and for our own families to persist forever.

(f.i) Conditions: exceptional personal righteousness, baptism, temple endowment.

(f.ii) Conditions: exceptional personal righteousness,  exceptional personal relationship with your spouse, baptism, temple marriage.

(g) is, perhaps, entirely unique.  One example was Adam's baptism, referenced above.  There are a handful of unique baptisms like his in scripture (another being Alma the elder).  In this role, the Holy Ghost is a unique "go-to" person to make a specific thing happen.  This is not an "achievable" condition.  You must be the right person, in the right place, at the right time, foreordained for the event.

Everything I have discussed happens by the "Power of the Holy Ghost," but I suspect that (3.f) is the most specific example of His unique power.

Please let me know if this addressed your question.  Thanks for giving me the chance to answer!