Ok, the pill is perfectly safe to use on it's own: there is a less than 1% chance of falling pregnant if you take it correctly. If you don't take it properly, it becomes less effective; so, if you do stop using condoms, you need to be careful, and if you do miss a pill, vomit within 3 hours of taking one etc, you MUST use condoms for 7 days and follow the instructions included with your pill.
So, in terms of safety re pregnancy, it's perfectly safe to just use the pill. Obviosuly condoms give even more protection, so if you're very worried it might be an idea to carry on using them; likewise, if you're bad at taking your pill it might be best to continue using them.
Second, if you are considering stopping using condoms, both yourself and your partner MUST have a check for STD's and other infections at your nearest GUM clinic (the Family Planning Association website has a list of clinics and you can search by postcode; otherwise your GP surgery or family planning clinic should be able to give you details). Even if you are sexually inexperienced, you should have this check, as non-sexually transmitted infections can also be passed on through sexual intercourse.
If the results are clear, then you can safely stop using condoms. However, I would then go for a follow-up STD check after 3-4 months to be on the safe side. This isn't a slight on your partners honesty, it's you taking responsibility for your health. IMPO it's a good idea to get into the habit of regular (12-18 months when in a long-term relationship; 6-12 months when not) STD checks, much as you would regularly visit your dentist or have your eyes checked.
So, in summary:
Pros of not using condoms IMPO are that it does increase sensation; I personally don't like using condoms and only use them when not in a relationship, at the start of a new relationshop before being tested, or when I'm not using an alternative method like the pill.
Cons of not using condoms are condoms DO give you extra protection against pregnancy, and they also protect against STD's, so if you don't use them regular screening is ESSENTIAL.
Finally, don't stop using condoms unless you are 100% comfortable with this.