How High?
After playing Kerbal Space Program with Realism Overhaul mod, I can tell you that a delta-V of around 9 km/s is needed to reach orbit (this is about 2 km/s more than orbital velocity; there are both gravity losses during the early, near vertical phase and significant air drag losses until the vehicle has exceeded approximately 100 km altitude). As noted in comments, any lesser amount than full orbital velocity will be suborbital, but the very same delta-V can cover a significant variation in range based on how much is directed upward vs. how much gets turned into horizontal velocity.
For instance (based on recent play in a Realistic Progression 1 career) I can tell you that 4 km/s delta-V can either push a rocket to around 500 km vertically and only make a few kilometers downrange distance, or send it nearly 1500 km downrange.
Therefore, there is no one answer to the dV requirement for an intercontinental flight (and as noted in comments, "intercontinental" needs better definition). The typical definition of an Intercontinental Ballistic Missile is one having more than 5 km/s dV with the design payload. This amount is sufficient to send the payload about 5000 km, perhaps a little more if launched due east from a tropical launch point.
Even the ~9 km/s needed to reach orbit when launching eastward from below about 40 degrees latitude will give a suborbital flight if you launch to the west, as you must subtract the Earth's ~500 m/s rotational velocity instead of adding it.
How long?
How long an intercontinental flight takes depends on multiple variables, too. With a minimum dV trajectory, it'll take longer than with a minimum-altitude (what the ballistic missile crowd call "depressed trajectory") path.
Minimum dV will give a burnout vector with a significant upward angle; for distances short enough to ignore the curvature of the Earth's surface, that's about 45 degrees (slightly lower if air drag is significant), with a lower angle the further you're casting your payload (because the Earth's surface "gets out of the way" the further you travel). This will throw your payload high, however; in the simple "flat Earth" case, about half the flight distance. If you're going a couple thousand kilometers, the time it takes to fall upward and back down for half that height is very substantial; where if you stay just out of the sensible atmosphere, more of your dV is expended horizontally and you gain both downrange velocity and some "weight loss" from getting closer to orbital speed.
In fact, if you're going to the antipodes (~12,000 km) you can get there in less than half the time (under an hour, to the limit of about 48 minutes if you actually establish an orbit and then break orbit to land) by staying as low as possible vs. saving a relatively small dV increment by flying minimum energy.
How hard?
The acceleration needed varies tremendously, too. The Space Shuttle used to throttle its engines to limit the G loads on the crew to no more than 3.0 G, while the suborbital Mercury flights pulled nearly 8 G at engine cutoff and more than that during reentry. The higher the G, the less gravity losses, which saves dV, but even highly trained and very fit humans aren't very functional at 8 G, while nearly anyone can remain functionally conscious and able to do low-stress tasks at 2.5 G. If you're launching a nuclear warhead, you can use much higher G loading than if you're launching live people, in other words, and you can save both dV and, by extension, transit time by doing so.
