The 5 bases you are referring to I am assuming are the -masu endings.
Eg. およぎます oyogimasu I swim
およぎません oyogimasen I don't swim
およぎました oyogimashita I swam
およぎませんでした oyogimasendeshita I did not swim.
およぎましょう oyogimashou Let us swim. (volitional)
You need to add the -masu -masen -mashita -masendeshita -mashou endings to the stem of the verb, which is obtained by dropping the -ru or by changing the last u--->i
Some phonetic changes might be necessary eg. tsu-->chi
In response to your question about politeness, you have polite and plain forms of verbs.
りんごをたべる？ "Ringo o taberu?" would be said between friends, where "taberu" is informal. Also called the dictionary form.
おちゃをのみますか？ "Ocha o nomimasu ka?" would be said between people not too familiar with one another. It's the -MASU ending which you'll see a lot.
めしあがってください！ ”Messhiagatte kudasai" which means "Please eat" is honorific,
It is composed of the -te form of "meshiagaru" the honorific version of "taberu" and the imperative of "kudasaru", the polite form of "kureru", to give.
The -ta form is the simple past tense, for example きのうは、ぎんこうにいった。
" Kinou wa, ginkou ni itta." Yesterday, I went to the bank" (plain).
However, the -masu ending are for polite speech. There are many other conjugations and endings for informal speech. A full list can be found here:
Replace (underscore) with "_". I can't seem to put underscores in web addresses when I post them.
ロケットジャパニーズで べんきょ たのしんで ください！