It's fine. It'd take quite a bit of oxygen to ruin a beer.
Without a recipe, I don't know how long you should keep it in secondary. Some recipes can be bottled straight from priimary; others need 6 months or more of aging. What kind of beer is this?
Yes, a hydrometer is absolutely essential. I didn't use one at first, even though one came with all the equipment I inherited, and I was really flying blind without it. Nobody here could help me much, because there were so many unanswerable questions. So, yeah, get that hydrometer quick.
68f might be a little warm for some yeast...especially when you factor in the added heat from fermentation activity.
You should always aerate, and always use a big yeast starter when using liquid yeast.