Very good question.
ASL does not have a standard writing system. From the perspective of using a system to document what you are learning, I would suggest that you use what is referred to as glossing. I would not recommend for example using Stokoe's system as it is too abstract for your purposes. Glossing is the representation of sign language in English print with certain rules governing its execution. For example, every sign has only one glossed representation. Although for your purposes this rule would not have to be followed 100 percent. As an example of this, if a linguist is using glossing for research purposes they would want to be more exact so that another person reading their glosses of expressions would replicate the exact signing. So depending on how exact you need to be would govern how many rules you apply to your glossing system. Most current day textbooks will explain the rules of the glossing systems they are using. Understand that the authors of these texts are creating glossing rules that will suit the purposes they have so there is not one universally standard way to gloss signs. You can look in the Signing Naturally curriculum (possibly having to look in the teacher edition of this curriculum, I'm not sure) and you will see an explanation of the glossing rules they are using. The ASL at Work curriculum explains the glossing rules the authors are using in the Student Text so students understand any glossing that may be occuring in the text.
I suggest you use the glossing rules that are used in whatever text you are using. Keep in mind that the glossing should be understandable to you primarily unless your teacher for some reason wants you to be glossing for class assignments and then you will have to conform to whatever glossing system your teacher is using.
Here is an example of how glossing my work to show you some of the general conventions.
English: Pam was very excited when she saw the dog.
ASL: P-A-M SEE DOG THERE, EXCITED-very SHE (user friendly glossing)
You have to know that indexing is being used and it is being glossed more freely to make it easier to understand the sentence and how to sign it. This would not be used by a researcher because "THERE" and "SHE" obviously are the same sign (index to third persons).
ASL with less user friendly more exact glossing.
P-A-M SEE DOG IX-left, EXCITED-very IX-right
The hypens in P-A-M indicate a fingerspelled name/word. The ALL CAPS as in SEE, DOG etc. indicate "a sign representation/gloss. IX indicates indexing to a location in space, EXCITED-very means that the sign is EXCITED and the "very" is showing how it is being modified.
Of course this is only one level of glossing because it is not showing the non-manual-signals on the face and body-movements that also are aspects that can be represented. Usually these are represented in a glossing system by drawing a line on top of the Glossed sentence and indicating what it happening like wh-q on the end of the line would indicate that a Wh question is being asked etc. But I think for your purposes this is getting a bit too complicated. You would have to decide at what level you wish to gloss sentences and the rules as I have stated previously that you want to apply.
For a wh-question sentence you could simply use for example,
ENGLISH - What is your name?
ASL: NAME YOU WHAT?
ENGLISH - It is not me.
ASL - ME-neg (head would be shaking as negatively as you sign "ME".
Hope this helps.
Bill