Rose - I started out the same - I still cant do the eliptical for more than 6 minutes and I too come off panting.
The treadmill - When I first joined the gym last Jan I hadnt run in years...the trainer suggested I try running for 30 seconds followed by 3 minutes of walking to get my breathe back etc....I even found the 30 seconds a struggle to start with

... I did this for a while then increased my run time to 35 then 40 seconds etc...I now run 1 minute and walk 1 minute... I can keep this up for about 30 minutes, then I walk witha hill gradient - usually about 6 or 8 for 15 mins, then if I have any energy left I do another 6 minutes run/walk, then a cool down walk.
I'm not fit at all and struggle with a lot of the equipment, but just persevere and you slowly get better. Keep at it girl
I also find that if I do a leg exercise, follow it up with eg arms - (rower), then stomach then back to legs - it gives that bit of your body time to recouperate before staring again
Some days are better than others too - some days I can only do 20 mins on the dreadmill, other times if I get the right track on the mp3 player I can do longer - music really helps me run to a beat.
Good luck Rose - everyone has to start somewhere.