Yes, as Northseaguy suggested, continue using a little lube around the rim if it helps and start off gently, not squeezing to the full extent. Do a few sessions of light suction, just to where it still feels comfortable, leaving them on for a few minutes each time, then releasing and letting them 'recover' for a few minutes more in between.
If the rims are sharp however, I do still recommend sending them in for a swap with a different brand. Lovehoney's customer care and returns policy are second to none, and there are several other brands of nipple suckers in their stock, as well as a nipple pump with different sized bands.
Once you feel confident, try squeezing more and getting to a point of suction that is only just bearable or slighty uncomfortable (you might even feel itchy), but not painful. As above, leave them for a few minutes or until they become painful/unbearable, whichever comes first, then letting them rest awhile before starting again. If you progress in this vein, you may find you can gradually take more suction or leave them on for a little longer.
You may choose to wait until you can take a full pump of the suckers before trying clamps, or you may want to incorporate them into your training.
Start out with a lightweight and highly adjustable type. I would suggest loops, tweezers, or bars. The size of your nipples will affect which type(s) work best for you. You may even be able to use them and the pump/suckers at the same time. (I think loops would be most compatible for that.)
http://www.lovehoney.co.uk/product.cfm?p=29732
http://www.lovehoney.co.uk/product.cfm?p=31536
http://www.lovehoney.co.uk/product.cfm?p=29731
Use the same technique as above; on for a few minutes, off for a few minutes. If you are interested in adding weights (or simply getting heavier clamps), you'll probably want to be sure you can take a hefty pinch, since obviously the heavier they are, the more likely they are to slip off without a good hold. Try tugging on your clamps to get used to the feeling/see if you like it/see how well your level of pinch holds. When you feel ready, I suggest graduating to a pair of adjustable alligator clamps or adjustable bite clamps.
I love these ones, but they are very strong and fairly heavy due to the double chain: http://www.lovehoney.co.uk/product.cfm?p=31503
I really want these ones, which are lightweight: http://www.lovehoney.co.uk/product.cfm?p=32927
and I have these on my list, too: http://www.lovehoney.co.uk/product.cfm?p=29741
Do NOT use any numbing agents or freeze your nipples before or during pumping/clamping sessions. You could do yourself a lot of unintentional harm or even permanent damage. If you experience soreness or irritation afterwards, you can soothe them when the session is over.
You CAN however use circulation stimulating balms/gels/creams/etcetera, like or similar to arousal balms. This may engorge the nipples more and further heighten sensitivity, so just be careful and use your good judgement.
There's a little bit of info in LH's guide too if you haven't checked it out already.
http://www.lovehoney.co.uk/bondage/buyers-guide/nipples-clamps-suckers-clips/