This is a bit difficult of an idea for me to articulate, so I'll try and do my best: If a character was to attempt and cast a spell with a casting time of 1 minute, while in combat, would they have to simply consume their action for 10 turns, but still move and take bonus actions? Would they be unable to do anything except cast that spell? Would they also be required to maintain concentration?
You have to concentrate to maintain the spell, and spend your action each turn casting the spell
Longer casting times are noted in the PHB and Basic rules. I'll quote the relevant text from D&D beyond (the official 5e web toolset), since it is the easiest source to search online.
Longer Casting Times
Certain spells (including spells cast as rituals) require more time to cast: minutes or even hours. When you cast a spell with a casting time longer than a single action or reaction, you must spend your action each turn casting the spell, and you must maintain your concentration while you do so. If your concentration is broken, the spell fails, but you don't expend a spell slot. If you want to try casting the spell again, you must start over.
You are still free to use your bonus action and reaction as appropriate
None of the rules for longer casting times here directly prevent you from using an available bonus action, reaction, or movement on your turn. So, you may use them as appropriate on your turn without your spell failing.
Using movement doesn't take an action, so you can still move and cast the spell
As noted in the PHB, basic rules, and D&D Beyond:
On your turn, you can move a distance up to your speed and take one action.
Moving on your turn doesn't take an action, and it doesn't break concentration. So unless the spell or some other effect requires you to stand still, you can still move on your turn.
You are limited as to casting spell with your bonus action though:
As noted in the rules on bonus action spells:
You can't cast another spell during the same turn, except for a cantrip with a casting time of 1 action.
Considering that you must continue to use your action each turn to cast the original spell, unless that spell is a cantrip, you wouldn't be able to cast a bonus action spell in the same turn. Luckily, spells that require your reaction don't have any such limits, so you can use those as often as you are able.