If you are keeping the same handset then you can simply ensure all contacts stored on the 'old' SIM card are copied to the handset before discarding the 'old' SIM. There will be a menu option for it somewhere. (The 'old' SIM is still readable by a handset even when it no longer connects to a network.)
If you are changing the handset too, then your contacts will either be stored on your 'old' handset or the 'old' SIM card, or some on each. Insert the 'old' SIM into the 'new' handset and find the menu option that allows you to copy or move contacts from SIM to handset.
Once you are sure the SIM stored contacts are now in your 'new' handset you can always clear the contacts from the 'old' SIM, reinsert it into your 'old' handset, then copy any handset stored contacts to the SIM.
After that you repeat the process to get all your contacts on your 'new' handset.
Once on your 'new' handset you have the option to copy them to the 'new' SIM if you wish.
Or you could purchase a SIM reader and store the information on your PC for copying anywhere you want, and for back-up purposes.