I work in sexual health. The sorts of infections that we would generally test for in asymptomatic people (healthy people who come in for a sexual health check up) are not the sort of things that could be spread by kissing. They are the sorts of things that are spread really only by unprotected oral/vaginal/anal sex. Herpes can be spread by kissing, but we wouldn't test for that and it's unlikely to be something you could catch from someone who didn't have an obvious cold sore on their lip.
I think when the relationship gets to the point that you are thinking about it becoming sexual, then that's a great time to talk about sexual health. Just say that you like to make sure you get tested between partners and you hope that your partner would too, ask if he'd be willing to get tested, and then say you'll share your results when they come in.
On a really practical level, these tests will only show you things that you could pass back and forth having condomless sex (e.g. chlamydia). They won't tell you about things that you can pass back and forth just by skin contact (e.g. herpes and HPV). You can find out if you have HPV by having your cervical smear as everyone gets tested now. We don't routinely test men though, similar with herpes. But at the very least, getting tested means you'll have a bit of reassurance if a condom were to break.
Your best bet though is to use a condom. Because an STI will give you an idea if either of you has an infection from awhile back. But it is still a snapshot in time. There is no guarantee that the results of a test from yesterday will still be accurate today - they could be having sex with someone else at the same time, or they could have tested in the window period and the infection was not yet detectable, but can still be transmitted.
STI testing is really important as part of just keeping up on your sexual health and if you are sexually active and concerned, definitely do it regularly. I think for women and heterosexual men, we recommend about every 6 months. But it isn't going to provide you with the certainty you are after. That comes with putting certain practices in place (using condoms), building trust in your relationship before having sex, and also accepting that you can't always control everything and there is always a bit of risk, even in a long-term relationship.