When I was an undergraduate we used Atkins and Jones' Chemical Principles: the Quest for Insight (link is to a slightly older edition because it's not a field whose basic principles have changed much in the last few years). If memory serves, it was pretty good. I'd also recommend checking out the MIT OCW site for 5.112 (that course will do a better job of preparing you for organic chemistry than 3.091, which is more materials focused).
It is certainly possible to start with an organic chemistry textbook as long as you have a good grasp of some fundamentals (what a chemical bond is, ionic vs. covalent bonding, electronegativity, bond dissociation energies, etc.) so if organic is really what you're after, feel free to give that a try. You can always pause that and go back to the general chemistry textbook if you feel like you're lacking foundation.
If memory serves, the Best Textbooks list has Clayden et al, which is the one that all the UK universities seem to use. It's a good textbook but if you're looking for an alternative suggestion, I used and liked Wade. Again, I'll put in a plug for OCW: 5.12 is the intro semester, with 5.13 following on.
Can I ask what the end goal for studying organic chemistry is? MCAT prep, learning biochemistry, and preparing to work in a synthesis lab all benefit from somewhat different approaches.