I have an entry level bike, Diamondback Response SE, that is a 21 speed. Is it possible to purchase and install a rear wheel with 10 speeds and change out the shifters and Derailuers or am I looking at frame mods too?

Thanks

Yes, but would be doing the math on cost of all the parts needed vs just buying a bike which is already 10 speed.

What's the objective of the upgrade? as you will still be left with an entry level bike, just with a nicer rear wheel & shifters

It's very possible, but not a very good return ratio on your $.
You "possibly" don't need a new hub.
Many of the later 7 speed bikes used an 8-10 speed free Hub with a spacer added to use up the space for the extra cog.

IF you upgrade, I'd stick to 9. The jump to 10 speed requires a noticeably more expensive/less durable chain. Shifters & cassettes also seem to take a disproportionate price increase when going above 9.
Cassette selection is very good between road & mountain.
I wouldn't bother with 8 speed.
The cost of shifters & cassette isn't really worth the cost for just 1 more gear.

Examine your freehub and see if there is a 4.5mm spacer between the largest cog and the hub. IF there is, you already have the wheel.

BTW- Possibly a simple cassette change can give you 7 gears that are usable. That's certainly as good as a cassette with more useless cogs on it.

Assuming you have a 12-32 cassette, maybe a 13-26 would work better? If you don't use the largest & smallest cogs on your existing cassette, you'd be more likely to use all the cogs on a 13-26. Maybe even a 13-23 if you don't have steep hills.

Why did you have to start another thread on this?

More speeds does not mean more speed or better performance. Learn how to ride and maintain your bike and know what you want to accomplish with any changes before jumping into spending money. As noted, in many cases it's better to sell your old bike and purchase a better new or used one than to "upgrade." There are many complications to making drive train changes.

Thanks very much for the good info. As I get more into riding I probably just upgrade. I've have been spending some time in local bike shops and dang, you can sink some money on these. A mountain bike that cost $6,000 is a tad on the extreme side. Around the $600 price can get you a very nice bike. I'm assuming of course. Like the hydraulic brakes.

I've done the 7 to 9 sp upgrade on a few older, good quality chrome-moly frames and it is both very doable and can be done at a reasonable cost if you stick to the entry to mid-level components (Alivio, Deore) and watch for good prices. 9sp parts are plentiful, durable and decently priced for the most part. If the parts you have are in good working order, I wouldn't upgrade just to upgrade, but if the original parts are nearing the end of their serviceable lives and you like the bike otherwise, an upgrade may be an alternative to replacing the whole bike.

As others have said, run the numbers to see how it compares to replacement of the bike and make sure what you end up with is going to be something you like. It's easy to get a lot of money into a mediocre bike.

Typically going 7 to 10 won't change the high or low gears, just give you more in between.

If you want more low gears, you can get Shimano Megarange which comes with a 34T cog.