Low quality steel is very cheap but also really heavy. You can get a whole crappy bike for a couple hundred bucks. Higher end steel isn't as cheap but is still relatively heavy (compared to aluminum or carbon). You can get this kind of bike in the $1000-2000 price range (e.g. Surly). Aluminum bikes tend to be inexpensive, but also not the lightest. These can also be priced at $1000-2000 (Specialized, Trek, Giant, ...). Carbon comes in a range of prices with various tradeoffs. You can get very budget carbon frames at like $1500 (Winspace) or whole bikes at like $3000 (Giant) (maybe $2000 in non-major brands, very discounted during current market conditions).

For titanium, I'm seeing Black Friday deals starting at like, $3200-3500 (Lynskey / Litespeed). But they're often sold at higher prices than carbon bikes. (For my money, I prefer carbon frames -- you get more flexibility in tube shapes and the end result can be lighter and stronger than titanium.)

Quality used bikes are remarkably affordable. I'm a sucker for 80s road bikes; $200 gets you something decent, bring it to a shop for a tune up and probably new brakes.

10-speeds are most common, so you have to be ok with that; although some vintage bikes did have 3 chain rings. Older road bikes tend to have narrow tires, because everybody thought they were faster; 1/4" tires are workable on unpaved mixed use trails, but 1/8" will be pushed around easily by tree roots and ruts.