Tell me please, what's the difference between these words and when can i use them?
This is a very interesting question：
V+了起来 Means the "V" has started or started at some time, maybe it still continues now. V+起来 can be used to mean "what will begin to happen" (usually in command mode).
他学了起来 (He began/started to learn, this means he is still learning).
Notice 1: We DON'T use V+了起来 to mean "what will begin to happen"
E.g:上课要迟到了，快跑起来（Hurry up, or you'll be late for class）
E.g: 上课要迟到了，他跑了起来（He began to hurry up because he would be late for class.）
Notice 2: V+了起来 cannot be used in a command mode
E.g: 天冷了，跑起来！(Run or you'll feel cold). We cannot say "天冷了，跑了起来！" at present.
2. For some verbs, as @Tang Ho♦ said, "V+起来" doesn't mean "start to do sth", but "up" instead (verbs with the meaning from low to high) or "do something harder"
E.g 1：站起来！(Stand up！）
E.g 2：坐起来！(Sit with your back straight！)
E.g 3：跑起来（It has two meanings：1：Start to run! 2.Run harder because you've almost stopped running）
3. For some other verbs, "V+起来" means "the appearance/mode/state of……", generally speaking, it is related to your mode, feeling...etc：
E.g：这东西吃起来怎么样？（How does it taste？）
E.g：他睡起来的样子真难看！（His sleeping posture looks uncomfortable.）
4. For some fixed phrase, "V+起来" cannot be split into "V"+"起来", it forms a whole to mean something special：
E.g: 你记起来这件事了么？“记起来” means “have you remembered？” or “想起来”
V + 起来 means start to do something, e.g: 体育老师对正在长跑的同学们说： 不要走，跑起来！（The P.E teacher said to the students who were running: Don't walk, start running!） In most situation, "V + 了起来" is similar with "V + 起来了" which means something just happenrd. e.g: 听到命令，他站了起来 and 听到命令，他站起来了(After hearing this order, he stood up).
[V + 起来] = [begin to + V]. It is simple enough,
One of the functions of 了 is to indicate the verb [is completed]. However, 起来 indicates the verb has just started. Therefore, [了] in [V + 了 + 起来] cannot be a verb particle that indicates the verb has completed.
Another function of 了 is to indicate [state changes]
But 了 is always placed at the end of the sentence when it is used for this function
我開始學英文了 = "I start to learn English" (state changes from not learning to learn)
我開始學了英文 = "I start to learned English". It is obviously incorrect because 開始 means "begin" to and 了 in this position would indicate [verb is completed]
了 can also be a modal word that only functions as an emphasis.
他突然笑了起來 (He suddenly laughed)
他突然笑起來 (He suddenly laughed)
Even without 了, the meaning remains the same
My conclusion: [了] in [V + 了 + 起来] is for emphasis
Sidenote: 起來 in 跳起來 (jump up) can be a directional particle that indicates 'V + upward' instead of a particle that indicates 'begin to + V'
V+了起来 = the action, described by the verb, happened just now. (This is the typical use of V+了)
我想了起来昨天發生的事 - I just now thought of an event that happened yesterday. (It wasn't in my mind before.)
他們吵了起來 - They (just now) started to quarrel.
他們一見面就聊了起來 - They immediately started to talk after they ran into each other.
V+起来 = act up the action prescribed by the verb, often a command, or demand.
- 站起来 -stand up; 坐起来 - sit up; 唱起来 - singing up (louder).
V+起来 = a phrase
說起來話長 - it is a long story (to tell/speak of).
提起來就氣 - it angers me even just to bring it up.
看起来不錯 - "from appearance, it does not look bad", which implies "it looks good".
You have a thing for 起来 I see!
I think you can best think of 起来 as a preposition. Sometimes 起来 means up, sometimes out, sometimes away and sometimes its own literal meaning, the context will decide. (Literally in 中国国歌：起来！起来！起来！）
1. 我总算硬撑着从床上爬了**起来**。 I managed to drag myself **out of** bed. 2. 我总算硬撑着从床上爬**起来**了。 I managed to drag myself **out of** bed. 3. 她胳膊给蜜蜂蜇了，肿了**起来**。 Her arm was beginning to swell **up** where the bee had stung her. 4. 她胳膊给蜜蜂蜇了，肿**起来**了。 Her arm was beginning to swell **up** where the bee had stung her. 5. 我躲了**起来**，不能冒险出去。 I stayed hidden **away**, I couldn't chance coming out. 6. 你躲**起来**，别出来冒风险。 You stay hidden **away**, don't take a risk coming out.
Prepositions often have interchangeable meanings, just look up the many meanings of "at" in a dictionary. The AHD has about 14 meanings for 'at'.
It is thought that "up" began in PIE （印欧语） with the meaning "under", then the meaning shifted to "from under away", then "away", then "over" and thus "up above".
爬起来: climb up, climb out, crawl up, crawl out (German: draufklettern, rausklettern)
肿起来: swell up (German: aufschwellen)
躲起来: hide away (German: sich verstecken, ver- has various meanings, and is not separable, but one is hinweg = away)