For example, let's examine character heaven. It has a Ren radical (men). And then we add a strip to become Tha (big). And then we add another strip to get heaven.
Now, is there a site to do so?
I think that'll be the easier for me to understand.
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The online chinese dictionary MDBG provides radical information for every character in its database. For instance, if you search for the character 天 (tiān) and click on the first result, the "Rad/Str" column reads 大 + 1, i.e., the radical 大 plus one stroke.
Zhongwen.com also gives information on character decomposition. The entry for 洋 reads "Water 水 with 羊 phonetic."
As Alenanno pointed out, not all parts of characters are radicals, but these two resources can give you more information about how characters are constructed.
The etymology part of the YellowBridge dictionary will show you this. Look up any character here, then click on etymology, and you will see the radicals in the 'etymology explorer'.
As mentioned in a question on searching by character parts, tatoeba.org has several tools related to character structure. The one you're interested in would be the one that explodes a character. It's not perfect, and your 天 example produced this:
一 大 大 一 大 大
However, 他 produced this:
亻 也 乜 丨 乚 乛 人
Another user here runs a site called HanziJS that does what you're asking for. (Sorry, I can't remember that user's name, but I discovered his website while reading here!)
You seem to assume that characters are only done by radicals, which is not true. Not all parts in a character are radicals.
Consider for example the character 他. The radical is 亻 which is 人, the other part is not a radical; its appearance is the same as the character for "also": 也, but this is not a radical. Radicals are very useful, since you can understand the probable "topic-area" of a character:
烊 has the radical 火 (fire), and means "melt"; 洋 has the radical 水 (water), and means ocean.
In the case of 天, the radical is the #37 in this page, which is 大 + another stroke (4 strokes in total).
If you want to learn characters by stroke, (for example for the character 天) you can use some sites that help you with animations/pictures showing the stroke order; you can also start from the wikipedia resources regarding stroke orders.
HanziCraft does the best job of breaking down characters into components (not just radicals) out of the sites I've tried, including the other answers here. It's not perfect (e.g. it doesn't do as great a job for traditional characters), but I've yet to find an alternative which provides as deep a level of component breakdown.
For example, 大:
Produces the following:
Decomposition: Once : 大 => 人, 一 Radical : 大 => 大 (big) Graphical : 大 => 人, 一
Everything is cross-referenced, so you can click on any component to break it down further e.g. if you click 人 from the above you get this:
Decomposition: Once : 人 => ㇒ Radical : 人 => 人 (human) Graphical : 人 => 人