I understand that ～もの converts the verb into a noun. So たべます meaning "to eat" becomes たべもの meaning "food". Can ～もの be applied to all verbs?
eg. ききます meaning "to listen" becomes ききもの meaning "music"?
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There is a way that ～もの can be applied to all verbs to "make them a noun", but it's not the way you're thinking of.
If you have a verb (e.g. 走る【はしる】 "to run") and a noun (e.g. 人【ひと】 "person"), you can always take the dictionary form (辞書形【じしょけい】) of the verb and put it before the noun, to get a construction that means something like "[noun] that [verb]s" (e.g. 走る人 "person who runs"). Since もの is a noun, you can do the same thing with any verb and もの as well.1 Note, however, that I put "make them a noun" above in quotes for a reason: this construction is not actually a noun. Rather, it forms what I guess you would call a "nominal" (thanks, @DariusJahandarie): a grammatical construction that behaves like noun, but isn't a noun (kind of like a noun phrase in English).
However, words like 食べもの【たべもの】 aren't formed this way. The way we can tell is that the dictionary form of 食べる【たべる】 is, well, 食べる, not 食べ. Instead, the word 食べもの is formed by taking the stem form of the verb and putting it before もの.2 This is not generalizable in the same way that the "dictionary form + noun" rule is. Only some verbs have a related noun of the form "stem form + もの". Some examples: 飲みもの【のみもの】 "drinks", 生きもの【いきもの】 "living things", 洗いもの【あらいもの】 "things to be washed [esp. clothes or dishes]".
聞く【きく】, however, is not one of those verbs - ききもの does not mean "music" or "things one listens to" or any such thing.3 Similarly, 走りもの doesn't mean anything at all, though you might think it would mean "runner" or something like that.
Incidentally, if you want to talk about "music", the right word to use is probably 音楽【おんがく】.
1 You should be careful about using "[dictionary form] + もの", though - there will often be a better choice. For example, you could say 食べるもの to mean "food", but that would be very strange, and people would look at you funny, and so you should stick to 食べもの instead.
2 When I say that 食べもの is "formed" in such-and-such a way, I am speaking anachronistically - in modern Japanese, 食べもの is a single noun, not a complex construction built from 食べ and もの. In other words, "[stem form] + [noun]" is not productive in modern Japanese, whereas "[dictionary form] + [noun]" is.
3 Okay, I lied for illustrative purposes. 聞き物 is actually a word, but it doesn't mean at all what you might expect it to mean based on an analogy with たべもの. Rather, it means "something worth listening to" (cf. the same in a monolingual dictionary). Furthermore, as @snailboat points out, 聞き物 is a very rare word - only two hits in the Balanced Corpus of Contemporary Written Japanese, versus over 10,000 for the more common word 音楽 (which, keep in mind, isn't a synonym). The moral of the story: be wary about "[stem form] + もの" nouns - not all the nouns that you imagine might exist actually do exist, and of the ones that do exist, some of them will not mean what you think they mean