"Watashi wa gakkou de tabemasu" means "I'm eating at school" or "I'm going to eat at school".
"Watashi wa gakkou de tabemashou" doesn't make sense as the "-shou" ending translates more or less to "let's ~verb". Thus, the sentence would translate to "I let's eat at school". Removing "watashi wa" (meaning "I") would make it "Let's eat at school".
"Desu" translates to "to be". It is the "is" or "are" verb used in most sentences. For example, "Watashi wa Mixi desu" meaning "I am Mixi". or "Kore wa isu desu" meaning "This is a chair".
It is also used by most young people to make sentences informal sentences polite. This is slang Japanese and is considered correct, but is not gramatically correct Japanese. It's like saying "I've gone to the store". It is accepted as correct English but the more grammatically correct form is "I have gone to the store".
An example of this is "Watashi wa gakkou ni iku desu" meaning "I am going to school". The informal/casual form is "Watashi wa gakkou ni iku" while the grammatically correct form is "Watashi wa gakkou ni ikimasu". By adding "desu" to the informal form it gives it a sense of politeness.
As for having to verbs in a sentence, this is possible. For example, "Watashi wa gakkou ni itte tabemasu" meaning "I will go to school and eat". This shows one action that follows another. "Iku" (to go) comes before "taberu" (to eat) to show the subject will "go" before "eating". The ~te form is used with the verbs that precede the final verb which is in infinitif form.
There is also ~nagara form which gives one verb being done at the same time as another. ie. "Watashi wa tabenagara gakkou ni ikimasu" meaning "I am going to school while eating".