There's a red lipstick out there for everyone, you just need to find what works for you best.
I would say start by working out your skin undertone. If you have a cool or rosy undertone, you will probably best suit a blue based red lipstick. If you are warmer, you may feel more comfortable wearing an orange or yellow based red (think coppery tones), or at least one without a strong blue undertone, though in my opinion blue based lipsticks are pretty universally flattering on most skin tones. If you are worried about your skin looking redder, invest in a good colour correcting primer (the Dr Jart Cicapair Colour Correcting Treatment works for me.)
Then work out what type of finish you'd be most comfortable with. Some red lipsticks are much higher maintenance than others. Generally, matte lipsticks will stay in place for longer but are much harder to touch up if you do smudge them, whereas sheer and satin lipsticks will transfer but are easily concealed/wiped away. Matte lipsticks tend to have a stronger pigment and provide the traditional 'bold' red look whereas satin and sheer can just provide a flattering wash of colour that will look incredibly natural.
Lip liners are a good idea if you are worried about bleeding. I would advise that you smudge the line a tiny bit so that it doesn't look too 'crisp' - this is where it can start to look a little clown like.
Some examples of blue based reds and their finishes:
MAC Ruby Woo - very matte and pigmented, the 'classic' red lip look
Nars Starwoman - very matte and pigmented, it's a liquid lipstick so needs time to dry down but once it does, it will generally stay put
MAC Lasting Passion - a more 'natural' matte, more comfortable to wear than regular matte, won't stay on as long and has a bit of transfer but won't generally bleed
YSL Rouge Volupte Shine in 45 (Rouge Tuxedo) - a glossier, sheerer finish. Can be built up to look more pigmented. Transfers but looks beautiful and flattering even on rougher, drier lips. Would wear with a lip liner.
Maybelline Red For Me - a cream finish, very pigmented. Nice and comfortable to wear but capable of transferring and bleeding so would probably need a lip liner.
Some examples of warmer reds:
MAC Chili - a cream finish but on the more matte end of cream. Very pigmented.
MAC Marrakesh - slightly more satin than cream, but still pigmented. I'd probably opt for a lip liner.
YSL Rouge Volupte Shine in 80 (Chili Tunique) - a glossy, sheerer finish, wear with a lip liner.
I haven't even touched on deeper, more burgundy reds - these can be spectacularly flattering but I'd advise to start with a light, bright red and then graduate to deeper ones if you feel confident!
I would generally advise going to a MAC or YSL counter and asking them to give you a good match based on your skin tone, 'boldness' level and longevity preferences. For me, MAC and YSL are THE lipstick brands due to their range of finishes.