I am still in the beginning stages of learning Japanese grammar and while I understand the meaning of the kanji I don't understand what のも means in the context of this sentence.
自分で言うのも なんだ けど
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See Schokolade's links for more information. Put together, 自分で言うのもなんだけど literally means "Saying it myself is a bit weird, but ...". It's a set phrase used before you want to say something you should not say. "I may not be the right person to say this, but ..." or "Though I say it myself, ...".
The 「の」is a foraml substantive in the sentence.
It mean 「のこと」、「のもの」、「のほう」, etc., making the attributive form have the same qualification as a substantive, but we generally don't say it completely for convenience.
Here 「も」is used to avoid expressing one's opinion directly, making a sentence mild.
The whole sentence means "I shouldn't / don't want to say this, but...".