Don't confuse parts of a sentence with parts of speech. Every word is a part of speech, either a noun, verb, pronoun, adjective, adverb, conjunction, article, preposition or interjection. It's important to learn what each of these means, and you'll need to know these to really understand parts of a sentence; just remember that the two terms are not the same.
The subject of a sentence refers to who or what is doing something. In "I want to watch TV," the subject is "I." The subject of a sentence can be a pronoun, like "I," or a noun, like "James." It may also be a phrase consisting of several parts of speech, such as "the big mean dog." In "the big mean dog bit me," "the" is an article, "big" and "mean" are adjectives" and "dog" is a noun, but "the big mean dog" is the subject.
Everything that is not the subject of a sentence is the predicate. In "the big mean dog bit me" the predicate is "bit me." The predicate always contains a verb, and may contain additional words. In "the big mean dog ran," "ran" is the predicate. The simplest sentences don't even have a subject. "Run!" is a complete sentence. The subject is the implied "you." The predicate is the verb "run!" which you better do, because there's a big mean dog around here somewhere.
A predicate can be broken down further. In "the big mean dog bit me," the predicate is "bit me." The verb is "bit" and "me" is the direct object. A direct object receives the action of the verb. Who did the biting? The subject did. The big mean dog. Who did he bite? The direct object, me. Notice that the direct object can be a pronoun, like me, a noun, like James, or a complex phrase like "the other big mean dog."
These are the basic parts of a sentence. Analyzing sentence structure can get much more complicated with compound sentences, predicate nominatives, gerunds and many other grammatical ideas. But get a handle on subject / verb/ direct object before you move forward. And watch out for big mean dogs.