Hi, Nolan! It's not so simple, you'd better ask seven separate questions.

Быть и являться have the same meaning , but they are used in different situations - Являться has more formal usage , like in official documents and сsientific literature . As well являться has the meaning - to appear . Явился - не запылился ( sarcastic saying , like they not glad to somebody's appearing, Гамлету являлся призрак его отца . the ghost of the father was appearing to Hamlet ) Видеть - to have ability to see , to observe Увидеть - to notice Посмотреть - to look Оставаться - umperfective aspect Остаться - perfective one ( like Perfect Tense in English ) Проводить и тратить are sort of different verbs , but they are close relating to time . the difference here is that проводить to send time , and тратить to wast time Написать - perfective aspect Писать - imperfective one Начать - perfective Начинать -imperfective Понять - perfective Понимать - imperfective

Other verbs are prefective/imperfective. It is a bit like "he talked", "he was talking" "he told/said" (perf). We don't have English tenses and it is hard for us to understand them. Instead we have perfective/imperfective aspect. Perfective aspect doesn't have present. because it almost always poits to a point (sometimes interval) in time, an _event_. I can go to school. I can be going to school. But "I said" is a single event. I know, you can say in English: "i'm saying", but in Russian it is unthinkable, you say "I'm speaking". Russian present maps to present (contiunous or not) in other languages. But Russian perfective past can be translated both with imperfect past, or with perfect tenses, depending on situation! Imp./perf. aspect distinguish between different things than perfect/imperfect tenses do. You will deal a LOT with this. My advise is: 1) if you want to imagine/feel it.... think of a 'point' in time. Я шел по дороге и встретил Юлю. I was walking down the road and met Yulia. "walking' is a process... "met" is an event. There is one exception to 'point' image: some по- verbs (perfective) mean "doing something _for_a_while_, often in lighmanded manner". 2) don't expect a simple rule to determine 'when it is more natural to use what'. Often both of them give THE SAME information:) If you went to .... yesterday, then you were walking to there. If you were walking, then you went:/ So we prefer one over another to express more shades. It is not about information. Yesterday, I was walking to... Yesterday, I went to... Seem to start two different stories, and draw my attention to different aspects. But information is the same. Sometimes it is as simple as.... In future, perfective si simple, while imperfective needs future of "to be". So imp.fut. is not too popular. We use it when we mean it.

Быть. The difference is stylistical. We don't have a 'copula' in Russian, that is "is". Nevertheless SOMETIMES we need it:) For emphasis. For this purpose we have these options: 1. "это". Definitions: Лингвистика - это наука, изучающая языки. And more colloquial: хххххх - это хорошо! xxxxxx, that's good! As you see, in writing I indicate omitted copula with a dash. In speech a dash is impossible, while a pause doesnt' sound good here...We use 'это' ("this"). And it means the same as "that's" in "that's good!". "That" somehow 'hints' on presence of " 's", and plays the role of a pause or a dash. This usage of "это" is somewhat... 'simple':) They don't use it in encyclopedias, but you can already find it in coursebooks' definitions) 2. является + instrumental case. Literally it means "appears" but can be used as 'is'. [.....] является самым высоким зданием западной Eвропы. Bookish and heavy. 3. when "is" is used for _placement_, like in "Boston is in USA" instead of является we say находится в... ....can be found in... 4. we may simply use "is":))) It is "есть". It is VERY serious, solemn and bookish style of definitions. More than является. On the other hand, it is funnier:) It is less long and it is conscious use of archaic language in modern context. Sort of, the question you discuss is so imortant that time boudaries can't matter. проводить - тратить проводить - to spend _time_. Use it with "time" or "summer" or words like that. Important thing about проводить Is HOW did you spend your time. Was it funny or were you playing poker or... тратить - to spend money. Also to spend time as a (valuable) resource. Important thing is - What For you spent it (when the time is fixed, like 'this evening') or - How Much of it did you spent (when the goal is known, like 'registration on Italki'). Sometimes it has a shade of 'waste' (because we don't have 'waste' besides slang expressions) - but not necessarily.

I agree with Igor and Denis. Studying imperfective/perfective will help with the most of the examples. And it's better to divide the question, since characters for the answers are limited.