How to make a table read/writes faster?
There are hundreds of answers.


Get rid of redundant indexes.  Index cookbook
Batch the INSERTs
Build "summary tables" (very useful in DW type apps).  Summary tables
Get a bigger AWS instance
With very rare exceptions Partitioning does not speed up anything.  Partition blog
Sharding is useful only if the data can be split is certain ways; meanwhile it slows down other operations.
AWS does a decent job of setting the tunable VARIABLES, so there is not likely to be any benefit in further tuning them.  Further analysis
Turn on the slowlog to find out which queries are the 'worst'.  Slowlog
If possible, avoid using GUIDs/UUIDs.  UUIDs don't scale
Don't do pagination via OFFSET.  Pagination blog


Please provide some clues of what the application is about.  Include SHOW CREATE TABLE and sample INSERTs and SELECTs.