The verb "отдыха́ть" is intransitive. You cannot "отдыхать что-то". Your sentence, when corrected, should read: "У́тром я обы́чно встаю́ и пробега́ю де́сять киломе́тров, пото́м ядаю́ отдохну́ть своим нога́м (let [my] feet rest).
About "бе́гаю" vs "бегу́" (not "вегу"): I would use "бегаю" if it's modified by time rather than distance ("встаю и полчаса бегаю"); with distance, the perfective "пробега́ю" seems more appropriate. Can't tell you why; it just sounds better.
Another stylistic comment: Russian uses pronouns much more sparingly than English, so you can safely get rid of the likes of "я" and "свои" in the second clause. That would improve its style.
no subject
ядаю́ отдохну́тьсвоимнога́м (let [my] feet rest).About "бе́гаю" vs "бегу́" (not "вегу"): I would use "бегаю" if it's modified by time rather than distance ("встаю и полчаса бегаю"); with distance, the perfective "пробега́ю" seems more appropriate. Can't tell you why; it just sounds better.
Another stylistic comment: Russian uses pronouns much more sparingly than English, so you can safely get rid of the likes of "я" and "свои" in the second clause. That would improve its style.