на ком, на чём?

"реке" is in педложный падеж, because in this case this word answers on the question "о чём? (about what?) - "о реке (about river)" "проводить" in this case means "to spend time"; " проводим" - present continuous of "проводить" I guess, In english this phrase sounds like "We are spending time at a river". And didn't understand question about винительный падеж... Sorry

There are two verbs in russian: the verb провожать (в прошедшем - проводить), that means accompany or see off and the verb проводить (в прошедшем - провести), that common use in the value "to spend time on smth". For example, "Я провожу её до метро" - "I see of her to the station" and "At the weekend I spend my time on my family" - "На выходных я провожу время с семьёй". By the way there is another value of verb проводить, that means "conduct" and usually is used to talk about electricity. For example "Металлы - хорошие проводники" - "Metals are good conductors" or "Metals conduct current good" - "Металлы хорошо проводят электрический ток".

So, prepositional case questions are Где? Where? O ком? About whom? О чём? About what? And here is some meanings, when we use the prepositional case: 1. The place of an action (где? Where?) + some verbs: быть (to be), жить (to live), находиться (to situated), работать (to work), учиться (to study), отдыхать (to rest), гулять (to walk). For a ex.: Она работает в школе. - She works at a school. Он учится в университете. He studies at the university. 2. The object of speech or thought (О ком? О чём? About whom? About what?) + verbs: говорить (to speak; to talk), рассказывать (to tell), думать (to think), спрашивать (to ask), писать (to write), мечтать (to dream). Ex.: Он мечтает о новой машине. He dreams about a new car. 3. When we talk about transport. (На чём? By which form of transport? How?). Ex.: Он ездит в офис на машине. He goes to the office by car. 4. Musical instruments. (На чём? Which instrument?). Ex: Он хорошо играет на гитаре. He plays the guitar very well. 5. Clothes. (В чём? In what?). Моя сестра в красном платье.- My sister in a red dress. 6. And we use Prepos. case, when we talk about time. (Когда? When?), (В каком году (in whiсh year), (В каком месяце (in which month), (На какой неделе (on which week). Ex.: Мой друг начал учить русский язык в прошлом году.- My friend started to learn Russian last year.

Here, the verb проводить means "to spend time", but you have forgotten the word "time"("Проводить время"). So "Мы проводим на реке" is not correct. It's better to say "Мы проводим время на реке"/ "Мы отдыхаем на реке". The most important thought in this sentence is what are we doing and where (где?). What - "мы проводим время", where - "на реке". The prepos.case is used to designate a place, so here, when we say "на реке" - a place where we spend time. And this case is always used with prepositions (“в” (in), “на” (on/at), when we say about a place, and “о” (about), when we say about a person (object) that is an object of speech and thought). It could be accus. case in this ex.: Я провожаю подругу в школу/в университет/ на концерт/ на работу/ в метро etc.