Softening a verb vs measuring quantity
一下 attaches to verbs to make the action feel brief, casual, or tentative. 看一下 is "take a quick look", 等一下 is "wait a moment", 问一下 is "ask quickly". 一点儿 attaches to verbs to mean "a little bit" of the action OR to adjectives to mean "a bit more". 吃一点儿 is "eat a little"; 便宜一点儿 is "a bit cheaper". The mental split: 一下 softens how; 一点儿 measures how much.
请等一下,我吃一点儿饭。
Qǐng děng yíxià, wǒ chī yìdiǎnr fàn.
Please wait a moment; I am eating a little food.
一下 = brief action
一下 is the politeness lubricant of Chinese. Every time you ask someone to do something, softening with 一下 makes the request feel lighter: 帮我看一下 (help me take a look), 等我一下 (wait for me a moment), 让我想一下 (let me think for a second). Drop the 一下 and the request becomes curt: 帮我看 sounds almost like a command. Native speakers use 一下 constantly in service and polite interactions.
一点儿 = small quantity with adjectives
With adjectives, 一点儿 goes after and means "a bit more in that direction": 大一点儿 (a bit bigger), 便宜一点儿 (a bit cheaper), 快一点儿 (a bit faster). This is the bargaining phrase, the request-a-size-change phrase, the adjust-the-volume phrase. Without 一点儿, the adjective feels absolute; with it, it feels like a small gradient change.
能再便宜一点儿吗?这个盘子能大一点儿的吗?
Néng zài piányi yìdiǎnr ma? Zhè ge pánzi néng dà yìdiǎnr de ma?
Can you make it a bit cheaper? Can this plate be a bit bigger?
一点儿 = small quantity with verbs
With verbs, 一点儿 means "a little bit of the action or thing": 吃一点儿 (eat a little), 学一点儿 (learn a little), 喝一点儿水 (drink a little water). This use requires 一点儿 to precede a noun or be embedded in a verb+一点儿 structure. Note that 一点儿 here is about quantity consumed or engaged, not about duration; for short duration use 一下.
Position and northern vs southern 儿化
一下 always follows the verb: 看一下 (look a moment), 想一下 (think a moment). 一点儿 follows the verb or adjective: 吃一点儿 (eat a little), 大一点儿 (a bit bigger). The 儿 on 一点儿 is classic northern Mandarin (Beijing especially); southerners and Taiwanese speakers often say 一点 without the 儿. Both are correct; keep consistent within your own speech.