Matching multiple columns on different data frames and getting other column as result

Use merge

df1 <- read.table(text='  chr    init
1  12  25289552
2   3 180418785
3   3 180434779', header=TRUE)


df2 <- read.table(text='    V1    V2     V3
10  1     69094 medium
11  1     69094 medium
12  12 25289552 high
13  1     69095 medium
14  3 180418785 medium
15  3 180434779 low', header=TRUE)


merge(df1, df2, by.x='init', by.y='V2') # this works!
       init chr V1     V3
1  25289552  12 12   high
2 180418785   3  3 medium
3 180434779   3  3    low

To get your desired output the way you show it

output <- merge(df1, df2, by.x='init', by.y='V2')[, c(2,1,4)]
colnames(output)[3] <- 'Mut' 
output
  chr      init    Mut
1  12  25289552   high
2   3 180418785 medium
3   3 180434779    low

df1 <- read.table(textConnection("   chr    init
 1  12  25289552
 2   3 180418785
 3   3 180434779"), header=T)

df2 <- read.table(textConnection("    V1    V2     V3
 10  1     69094 medium
 11  1     69094 medium
 12  12 25289552 high
 13  1     69095 medium
 14  3 180418785 medium
 15  3 180434779 low"), header=T)

# You have to select the values of df2$V3 such as their corresponding V2 
# are equal to the values of df1$init
df1$Mut <- df2$V3[ df2$V2 %in% df1$init]

df1
  chr      init    Mut
1  12  25289552   high
2   3 180418785 medium
3   3 180434779    low