Before you are even hired, share high level budget numbers from past projects to reference with your prospective client. You don't have to get into the weeds by contrasting material prices, for instance, at this stage - you will want to discuss more impactful features such as major movement of HVAC, penetrations or movement of structural walls, changes to the building envelope, or multi-floor impacts such as a new elevator location. Be very clear that these are generalized prices for initial reference only, but if the client's expectations for budget are unreasonable, that will quickly become clear, and they will be able to either reassess project finances or narrow priorities before the project moves forward.
At the first design meeting, broadly rank large scope items by how much structural change they entail, how many trades are involved, and whether they are conditional on other areas of scope being included. For instance, a project relocating a staircase will involve structural work on multiple floors, extensive permitting, and the need to, at a minimum, refinish all the sites touching the old location where disruption has occurred. Often when there are budget constraints, a choice is needed between the large, structurally impactful desired outcome and a package of smaller, more decorative areas. These could be visually more impactful than a large structural change, but the client needs to have an early awareness of the investments associated.
Your next objective when discussing these possibilities with the client should be to distinguish items which can be achieved at a variety of price points (such as material selection) from items with established costs (such as re-wiring labor). They may or may not be willing to compromise on the quality of finishes and fixtures to cover more scope - and you should be clear about your own standards as well, and give them honest feedback about which lines you draw in material quality - but there are other costs which are inflexible and which will be prohibitive based on the budget given. These are often labor-related, but necessary permitting, site work, or specialty consulting are also factors that may be cost prohibitive.
How many design packages you present to the client will be connected to your proposal terms. One of the many things you will need to discuss before a contract is signed is how multiple designs or changes to the design are accounted for (after all, you deserve to be paid for your time). Some clients will be able to easily self-identify as very decisive and/or receptive to designer recommendations, or as someone that needs more choices or more time. But for everyone else, it is wise to assume the design process will not be completely straightforward, and to plan time for multiple design variations into your proposal. If the client pushes back you can make adjustments, but at least there will be a mutual understanding of what is included.
Contingencies, an important piece of any contract, can be difficult for clients to rationalize or take seriously. Based on past projections vs actuals for similar situations, you can get a fairly logical sense for where contingencies are most needed and build them into the actual line items. This is almost synonymous with conservative budgeting, except that the level of conservatism varies based on your past data. Your database is an invaluable tool for estimation - make sure to keep detailed records for each project type and each contractor you work with to achieve the most accurate predictions for each budget you share with prospective clients.
Last but not least, have a plan to cover yourself from client-led delays to the project. Not all clients will respect the time of the designer and/or contractors when continuously changing their mind mid-project, but they are much more likely to respect the cost. At the very beginning, when you're discussing your contract, clearly articulate your late fee structure (which you should definitely have to protect yourself). Then, when the project goes to bid, at the contract meeting with the GC and client, discuss again the fees incurred for client-induced delays. There is likely a weekly general conditions charge they will incur for each extra week the GC has to be mobilized due to extra work from the client.