You can change the suspend char using stty. In your case, for example, you can disable the "suspend" ability with

stty susp undef 

or change it to for example ctrl-E with

stty susp '^E' 

and you can check it:

[romano:~/Downloads/mps] % sleep 1000 
^E
[1]  + 2681 suspended  sleep 1000
1z [romano:~/Downloads/mps] 20 % fg
[1]  + 2681 continued  sleep 1000
^Z 

stty -a gives you all the special chars:

[romano:~/Downloads/mps] % stty -a
speed 38400 baud; rows 24; columns 114; line = 0;
intr = ^C; quit = ^\; erase = ^?; kill = ^U; eof = ^D; eol = M-^?; eol2 = M-^?; swtch = M-^?; start = ^Q;
stop = ^S; susp = ^E; rprnt = ^R; werase = ^W; lnext = ^V; flush = ^O; min = 1; time = 0;
-parenb -parodd cs8 hupcl -cstopb cread -clocal -crtscts
-ignbrk brkint -ignpar -parmrk -inpck -istrip -inlcr -igncr icrnl ixon -ixoff -iuclc ixany imaxbel iutf8
opost -olcuc -ocrnl onlcr -onocr -onlret -ofill -ofdel nl0 cr0 tab0 bs0 vt0 ff0
isig icanon iexten echo echoe echok -echonl -noflsh -xcase -tostop -echoprt echoctl echoke

... although after re-reading the question you will probably greatly benefit from a terminal multiplexer, as @Shaun suggested. My personal favorite is the good old screen, which has a lot of tutorial around, even in the written form which I prefer to videos...