I showed few tactics to simplify and visualise data in R using tidyverse You can explore more by taking this as start point.

library(tidyverse)
ss <- df %>% 
as_tibble() %>% 
gather(key = "cond" , value = "value" , -AGI) %>%  ## wide to long format
separate(cond , into = c("Genotype", "Time", "Replicate") , sep = "_+") ## separate each attribute 


# A tibble: 108 x 5
   AGI       Genotype Time  Replicate value
   <chr>     <chr>    <chr> <chr>     <dbl>
 1 ATCG01240 aox2     0h    1          15.8
 2 ATCG01310 aox2     0h    1          14.8
 3 ATMG00070 aox2     0h    1          14.7
 4 ATCG01240 aox2     0h    2          16.1
 5 ATCG01310 aox2     0h    2          14.5
 6 ATMG00070 aox2     0h    2          14.5
 7 ATCG01240 aox2     0h    3          14.6
 8 ATCG01310 aox2     0h    3          14.7
 9 ATMG00070 aox2     0h    3          13.1
10 ATCG01240 aox2     0h    4          15.7
# … with 98 more rows


## average out replicates 
ss_m <- ss %>% group_by(AGI, Genotype , Time) %>% summarise(replicates_mean = mean(value))

ss_m
# A tibble: 27 x 4
# Groups:   AGI, Genotype [?]
   AGI       Genotype Time  replicates_mean
   <chr>     <chr>    <chr>           <dbl>
 1 ATCG01240 aox2     0h               15.6
 2 ATCG01240 aox2     12h              14.9
 3 ATCG01240 aox2     24h              15.2
 4 ATCG01240 aox5     0h               15.2
 5 ATCG01240 aox5     12h              14.6
 6 ATCG01240 aox5     24h              15.3
 7 ATCG01240 WT       0h               14.2
 8 ATCG01240 WT       12h              14.8
 9 ATCG01240 WT       24h              15.3
10 ATCG01310 aox2     0h               14.8
# … with 17 more rows

comparing timepoint

bplot <- ss_m %>% ggplot() + geom_boxplot(aes(x = Time, y = replicates_mean , fill = Genotype)) +  theme_bw() + theme(text = element_text(size = 20))
ggsave(filename = "boxplot.png" ,plot = bplot)

comparing genotype

bplot2 <- ss_m %>% ggplot() + geom_boxplot(aes(x = Genotype, y = replicates_mean , fill = Time)) +  theme_bw() + theme(text = element_text(size = 20))
ggsave(filename = "boxplot2.png" ,plot = bplot2)

======================Update ===============================

Convert the data in to the format you asked. ( std dev and mean only)

ss_mm <- ss %>% group_by(AGI, Genotype , Time) %>% 
        summarise(replicates_mean = mean(value) , stddev = sd(value)) %>% ## add stddev and mean 
        unite(Genotype, Time , col = "Genotype_Time" , sep = "_") %>% ## unite genotype and time in a single column
        gather(key = summary_type , value = value , replicates_mean , stddev) %>% ## create summary_type variable 
        unite(Genotype_Time, summary_type , col = "Genotype_Time_summary_type",sep = "_") %>% ##create Genotype_Time_summary_type variable
        spread(Genotype_Time_summary_type , value) ## wide format 

## summary of final table. 
glimpse(ss_mm)

Observations: 3
Variables: 19
Groups: AGI [3]
$ AGI                      <chr> "ATCG01240", "ATCG01310", "ATMG00070"
$ aox2_0h_replicates_mean  <dbl> 15.55477, 14.80490, 14.65538
$ aox2_0h_stddev           <dbl> 0.6240735, 0.2689779, 1.3299868
$ aox2_12h_replicates_mean <dbl> 14.88098, 15.01747, 14.93024
$ aox2_12h_stddev          <dbl> 0.42239203, 0.09092439, 0.60146632
$ aox2_24h_replicates_mean <dbl> 15.16378, 14.88365, 14.29683
$ aox2_24h_stddev          <dbl> 0.33059885, 0.07035039, 0.90767009
$ aox5_0h_replicates_mean  <dbl> 15.18685, 14.50503, 15.07261
$ aox5_0h_stddev           <dbl> 0.7175443, 0.7168420, 0.8547323
$ aox5_12h_replicates_mean <dbl> 14.57651, 15.05366, 14.96086
$ aox5_12h_stddev          <dbl> 0.07459644, 0.16231378, 1.28919700
$ aox5_24h_replicates_mean <dbl> 15.32162, 15.03121, 14.77809
$ aox5_24h_stddev          <dbl> 0.5483318, 0.1643006, 0.7481768
$ WT_0h_replicates_mean    <dbl> 14.19501, 14.09748, 15.02218
$ WT_0h_stddev             <dbl> 0.4794059, 1.2639163, 0.1382836
$ WT_12h_replicates_mean   <dbl> 14.84365, 14.16882, 14.69625
$ WT_12h_stddev            <dbl> 0.1521183, 0.8097963, 1.2471750
$ WT_24h_replicates_mean   <dbl> 15.32586, 14.31087, 14.92495
$ WT_24h_stddev            <dbl> 0.5509899, 0.7280381, 1.5358987