I'm pretty interested in buying the D&D Starter Set as a way to get introduced to D&D and being a GM/DM for the first time. (more context in this other question I have asked, regarding the expected length of this campaign)

After finishing the starter set campaign, can I make my party "move" to another bigger module easily (using the same character sheets)?

I found this question which is related, but doesn't really answer if it is possible to make a transition with the same characters instead of just starting a new game.

I'd like answers specifically for this campaign as it's a basic one, so there shouldn't be any "compatibility" problems this way. Answers from DMs who've already tried doing this would be greatly appreciated.

Yes, you can reuse characters.

There is nothing particular or special to Lost Mines of Phandelver that requires handling differently from any other content. Characters are not implicitly one-and-done - you don't throw away the character when you finish the book. Just because many of the hardcover adventures are designed to start at L1, doesn't mean they must do so.

Some modules even specifically call out how to blend them. For example, Storm King's Thunder has a section in the appendices that covers what to do with characters coming out of Lost Mines of Phandelver. The short version is "skip chapter one and start with chapter two", but there are a few more specific suggestions of how to work it into a story. Other modules have different hooks, or leave you room to develop your own hooks.

Balancing Being Overleveled

If you're starting a hardcover adventure with characters above the recommended level, you'll need to adjust for that (or at least be aware of it). Early encounters will certainly need the opposition to be buffed up in some way, and even later encounters may need it (the characters are starting overleveled, so they may be overleveled at the end, too). That's where DM discretion comes in.

This is also an opportunity to take into account player preference. Some players like the edge-of-your-seat danger and risk of hard or deadly encounters, and the tactical thought that goes into swinging the odds in your favor to win. Other players prefer waltzing into a room and slaughtering enemies by the score, making each encounter a blood-splattered mess. It's up to you as the DM to gauge your players and adjust the encounters to suit.

Milestone Advancement

One of the easier ways to control character level is to not award experience points, but to assign additional levels as the story demands it. This is known as milestone advancement. This can relieve some of the burden of a party being overleveled, because you simply don't have them level up from encounters that aren't really a challenge.

Yes, you can

Several, if not all, official adventures offer several hooks and at least one of them usually is suited to tie in from the starter set. When the PCs join a faction during LMOP it is even easier.

Yes, absolutely!

One published module this works exceptionally well with is Storm King's Thunder which (after a short "prequel" that isn't terribly relevant) can begin in one of three locations on the Sword Coast (the region Lost Mines takes place in). One of them is Triboar which is very close to Phandalin (you may recall that it lies on the aptly Triboar Trail).

Storm King's Thunder even specifically recommends that it can be used right after Lost Mines of Phandelver in the way that I suggest above. It says to skip the "prequel" when doing this. It also describes the "prequel" as being skippable if you want to start at a higher level instead of level 1, so you really shouldn't be missing too much.