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By dimensions, i assume you mean, [M], [L], [T], examples like that.
you don't need to memorise dimensions of speed or force or anything. you just need to remember their units(si or cgs), then, you can write their dimensions.
for example, speed's si unit is ms^(-1).
m indicates metre for distance, can be denoted by[L], s denotes seconds for time, can be denoted by [T].
so, dimensions for speed is written as [L][T]^(-1).
SIMILARLY, for force, units are kgms^-1.
so, its dimensions are [M][L][T]^-1.
keep practicing questions. eventually you will memorise the units. once that's done, no more worrying about dimensions.